INFSCI 2595 Fall 2023 Homework: 08

Assigned October 27, 2023; Due: November 3, 2023

Collaborators

Include the names of your collaborators here.

Overview

This homework assignment reviews important aspects of training linear models focusing on the relationship between complexity and model performance. You will fit non-Bayesian and Bayesian linear models of varying levels of complexity. You will compare their performance and examine their predictive trends. Unlike earlier assignments you will examine both types of uncertainty in addition to the trend. This way you gain experience with the relationship between complexity, training set performance, confidence intervals, and prediction intervals.

This assignment also requires you to fit the Bayesian models with various prior standard deviations. In this way, you learn about the prior’s role on coefficient estimates and predictive trends. These tasks introduce to the concept that the prior regularizes coefficients.

Lastly, you are introduced to non-Bayesian regularization with Lasso regression via the glmnet package. If you do not have glmnet installed PLEASE download it before starting the assignment.

IMPORTANT: The RMarkdown assumes you have downloaded the data set (CSV file) to the same directory you saved the template Rmarkdown file. If you do not have the CSV files in the correct location, the data will not be loaded correctly.

IMPORTANT!!!

Certain code chunks are created for you. Each code chunk has eval=TRUE set in the chunk options. You MUST change it to be eval=TRUE in order for the code chunks to be evaluated when rendering the document.

You are free to add more code chunks if you would like.

Load packages

This assignment will use packages from the tidyverse suite as well as the coefplot package. Those packages are imported for you below.

library(tidyverse)

## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.1.2     ✔ readr     2.1.4
## ✔ forcats   1.0.0     ✔ stringr   1.5.0
## ✔ ggplot2   3.4.3     ✔ tibble    3.2.1
## ✔ lubridate 1.9.2     ✔ tidyr     1.3.0
## ✔ purrr     1.0.2     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

library(coefplot)

This assignment also uses the splines and MASS packages. Both are installed with base R and so you do not need to download any additional packages to complete the assignment.

The last question in the assignment uses the glmnet package. As stated previously, please download and install glmnet if you do not currently have it.

Problem 01

You will fit and compare 6 models of varying complexity using non-Bayesian methods. The unknown parameters will be be estimated by finding their Maximum Likelihood Estimates (MLE). You are allowed to use the lm() function for this problem.

The data are loaded in the code chunk and a glimpse is shown for you below. There are 2 continuous inputs, x1 and x2, and a continuous response y.

hw_file_path <- 'hw08_data.csv'

df <- readr::read_csv(hw_file_path, col_names = TRUE)

## Rows: 100 Columns: 3
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (3): x1, x2, y
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

df %>% glimpse()

## Rows: 100
## Columns: 3
## $ x1 <dbl> -0.30923281, 0.63127211, -0.68276690, 0.26930562, 0.37252021, 1.296…
## $ x2 <dbl> 0.308779853, -0.547919793, 2.166449412, 1.209703658, 0.785485991, -…
## $ y  <dbl> 0.43636596, 1.37562976, -0.84366730, -0.43080811, 0.77456951, 1.361…

1a)

Create a scatter plot between the response, y, and each input using ggplot().

Based on the visualizations, do you think there are trends between either input and the response?

SOLUTION

df %>%
  ggplot(mapping = aes(y = y)) +
  geom_point(mapping = aes(x = x1, color = 'red')) +
  geom_point(mapping = aes(x = x2, color = 'blue'))

1b)

You will fit multiple models of varying complexity in this problem. You will start with linear additive features which add the effect of one input with the other. Your model therefore controls for both inputs.

Fit a model with linear additive features to predict the response, y. Use the formula interface and the lm() function to fit the model. Assign the result to the mod01 object.

Visualize the coefficient summaries with the coefplot() function. Are any of the features statistically significant?

SOLUTION

### add more code chunks if you like
mod01 <- lm(y ~ x1 + x2, data = df)
coefplot(mod01)

1c)

As discussed in lecture, we can derive features from inputs. We have worked with polynomial features and spline-based features in previous assignments. Features can also be derived as the products between different inputs. A feature calculated as the product of multiple inputs is usually referred to as the interaction between those inputs.

In the formula interface, a product of two inputs is denoted by the :. And so if we want to include just the multiplication of x1 and x2 in a model we would type, x1:x2. We can then include main-effect terms by including the additive features within the formula. Thus, the formula for a model with additive features and the interaction between x1 and x2 is:

y ~ x1 + x2 + x1:x2

However, the formula interface provides a short-cut to create main effects and interaction features. In the formula interface, the * operator will generate all main-effects and all interactions for us.

Fit a model with all main-effect and all-interaction features between x1 and x2 using the short-cut * operator within the formula interface. Assign the result to the mod02 object.

Visualize the coefficient summaries with the coefplot() function. How many features are present in the model? Are any of the features statistically significant?

SOLUTION

### add more code chunks if you like
mod02 <- lm(y ~ x1*x2, data = df)
coefplot(mod02)

1d)

The * operator will interact more than just inputs. We can interact expressions or groups of features together. To interact one group of features by another group of features, we just need to enclose each group within parenthesis, (), and separate them by the * operator. The line of code below shows how this works with the <expression 1> and <expression 2> as place holders for any expression we want to use.

(<expression 1>) * (<expression 2>)

Fit a model which interacts linear and quadratic features from x1 with linear and quadratic features from x2. Assign the result to the mod03 object.

Visualize the coefficient summaries with the coefplot() function. How many features are present in the model? Are any of the features statistically significant?

HINT: Remember to use the I() function when typing polynomials in the formula interface.

SOLUTION

### add more code chunks if you like
mod03 <- lm(y ~ (x1 + I(x1^2))*(x2 + I(x2^2)), data = df)
coefplot(mod03)

1e)

Let’s now try a more complicated model.

Fit a model which interacts linear, quadratic, cubic, and quartic (4th degree) polynomial features from x1 with linear, quadratic, cubic, and quartic (4th degree) polynomial features from x2. Assign the result to the mod04 object.

Visualize the coefficient summaries with the coefplot() function. Are any of the features statistically significant?

SOLUTION

### add more code chunks if you like
mod04 <- lm(y ~ (x1 + I(x1^2) + I(x1^3) + I(x1^4)) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), data = df)
coefplot(mod04)

1f)

Let’s try using spline based features. We will use a high degree-of-freedom natural spline applied to x1 and interact those features with polynomial features derived from x2.

Fit a model which interacts a 12 degree-of-freedom natural (DOF) spline from x1 with linear and quadrtic polyonomial features from x2. Assign the result to mod05.

Visualize the coefficient summaries with the coefplot() function. Are any of the features statistically significant?

SOLUTION

### add more code chunks if you like
mod05 <- lm(y ~ splines::ns(x1, df = 12) * I(x2^2), data = df)
coefplot(mod05)

1g)

Let’s fit one final model.

Fit a model which interacts a 12 degree-of-freedom natural spline from x1 with linear, quadrtic, cubic, and quartic (4th degree) polyonomial features from x2. Assign the result to mod05.

Visualize the coefficient summaries with the coefplot() function. Are any of the features statistically significant?

SOLUTION

### add more code chunks if you like
mod06 <- lm(y ~ splines::ns(x1, df = 12) * (x2 +I(x2^2) + I(x2^3) + I(x2^4)), data = df)
coefplot(mod06)

1h)

Now that you have fit multiple models of varying complexity, it is time to identify the best performing model.

Identify the best model considering training set only performance metrics. Which model is best according to R-squared? Which model is best according to AIC? Which model is best according to BIC?

HINT: The brooom::glance() function can be helpful here. The broom package is installed with tidyverse and so you should have it already.

SOLUTION

### add more code chunks if you like
models_glance <- bind_rows(
  broom::glance(mod01),
  broom::glance(mod02),
  broom::glance(mod03),
  broom::glance(mod04),
  broom::glance(mod05),
  broom::glance(mod06),
)
models_glance

## # A tibble: 6 × 12
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <dbl>  <dbl> <dbl> <dbl>
## 1    0.0594        0.0401 0.980      3.07 5.12e- 2     2 -138.   285.  295.
## 2    0.113         0.0853 0.956      4.08 9.00e- 3     3 -135.   281.  294.
## 3    0.547         0.507  0.702     13.7  7.25e-13     8 -102.   224.  250.
## 4    0.599         0.470  0.728      4.66 1.51e- 7    24  -95.7  243.  311.
## 5    0.647         0.527  0.688      5.42 7.03e- 9    25  -89.4  233.  303.
## 6    0.782         0.383  0.785      1.96 1.64e- 2    64  -65.3  263.  434.
## # ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>

sprintf('mod0%d', which.max(models_glance$r.squared))

## [1] "mod06"

sprintf('mod0%d', which.min(models_glance$AIC))

## [1] "mod03"

Problem 02

Now that you know which model is best, let’s visualize the predictive trends from the six models. This will help us better understand their performance and behavior.

2a)

You will define a prediction or visualization test grid. This grid will allow you to visualize behavior with respect to x1 for multiple values of x2.

Create a grid of input values where x1 consists of 101 evenly spaced points between -3.2 and 3.2 and x2 is 9 evenly spaced points between -3 and 3. The expand.grid() function is started for you and the data type conversion is provided to force the result to be a tibble.

SOLUTION

?seq
viz_grid <- expand.grid(x1 = seq(-3.2, 3.2, length.out = 101),
                        x2 = seq(-3, 3, length.out = 9),
                        KEEP.OUT.ATTRS = FALSE,
                        stringsAsFactors = FALSE) %>% 
  as.data.frame() %>% tibble::as_tibble()

2b)

You will make predictions for each of the models and visualize their trends. A function, tidy_predict(), is created for you which assembles the predicted mean trend, the confidence interval, and the prediction interval into a tibble for you. The result include the input values to streamline making the visualizations.

tidy_predict <- function(mod, xnew)
{
  pred_df <- predict(mod, xnew, interval = "confidence") %>% 
    as.data.frame() %>% tibble::as_tibble() %>% 
    dplyr::select(pred = fit, ci_lwr = lwr, ci_upr = upr) %>% 
    bind_cols(predict(mod, xnew, interval = 'prediction') %>% 
                as.data.frame() %>% tibble::as_tibble() %>% 
                dplyr::select(pred_lwr = lwr, pred_upr = upr))
  
  xnew %>% bind_cols(pred_df)
}

The first argument to the tidy_predict() function is a lm() model object and the second argument is new or test dataframe of inputs. When working with lm() and its predict() method, the functions will create the test design matrix consistent with the training design basis. It does so via the model object’s formula which is contained within the lm() model object. The lm() object therefore takes care of the heavy lifting for us!

Make predictions with each of the six models you fit in Problem 01 using the visualization grid, viz_grid. The predictions should be assigned to the variables pred_lm_01 through pred_lm_06 where the number is consistent with the model number fit previously.

SOLUTION

pred_lm_01 <- tidy_predict(mod01, viz_grid)

pred_lm_02 <- tidy_predict(mod02, viz_grid)

pred_lm_03 <- tidy_predict(mod03, viz_grid)

pred_lm_04 <- tidy_predict(mod04, viz_grid)

pred_lm_05 <- tidy_predict(mod05, viz_grid)

pred_lm_06 <- tidy_predict(mod06, viz_grid)

2c)

You will now visualize the predictive trends and the confidence and prediction intervals for each model. The pred column in of each pred_lm_ objects is the predictive mean trend. The ci_lwr and ci_upr columns are the lower and upper bounds of the confidence interval, respectively. The pred_lwr and pred_upr columns are the lower and upper bounds of the prediction interval, respectively.

You will use ggplot() to visualize the predictions. You will use geom_line() to visualize the mean trend and geom_ribbon() to visualize the uncertainty intervals.

Visualize the predictions of each model on the visualization grid. Pipe the pred_lm_ object to ggplot() and map the x1 variable to the x-aesthetic. Add three geometric object layers. The first and second layers are each geom_ribbon() and the third layer is geom_line(). In the geom_line() layer map the pred variable to the y aesthetic. In the first geom_ribbon() layer, map pred_lwr and pred_upr to the ymin and ymax aesthetics, respectively. Hard code the fill to be orange in the first geom_ribbon() layer (outside the aes() call). In the second geom_ribbon() layer, map ci_lwr and ci_upr to the ymin and ymax aesthetics, respectively. Hard code the fill to be grey in the second geom_ribbon() layer (outside the aes() call). Include facet_wrap() with the facets with controlled by the x2 variable.

To help compare the visualizations across models include a coord_cartesian() layer with the ylim argument set to c(-7,7).

Each model’s prediction visualization should be created in a separate code chunk.

SOLUTION

Create separate code chunks for each visualization.

pred_lm_01 %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
  geom_line(mapping = aes(y = pred)) + 
  facet_wrap(~ x2)

pred_lm_02 %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
  geom_line(mapping = aes(y = pred)) + 
  facet_wrap(~ x2)

pred_lm_03 %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
  geom_line(mapping = aes(y = pred)) + 
  facet_wrap(~ x2)

pred_lm_04 %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
  geom_line(mapping = aes(y = pred)) + 
  facet_wrap(~ x2)

pred_lm_05 %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
  geom_line(mapping = aes(y = pred)) + 
  facet_wrap(~ x2)

pred_lm_06 %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
  geom_line(mapping = aes(y = pred)) + 
  facet_wrap(~ x2)

2d)

Do you feel the predictions are consistent with the model performance rankings based on AIC/BIC? What is the defining characteristic of the models considered to be the worst by AIC/BIC?

SOLUTION

What do you think?

AIC considers the number of parameters as a penalty. Yes, I feel the rankings according to AIC are justified because the we see the scale of confidence interval is very high for the complex model, which means that the models have no idea where the answer lies.

Problem 03

Now that you have fit non-Bayesian linear models with maximum likelihood estimation, it is time to use Bayesian models to understand the influence of the prior on the model behavior.

Regardless of your answers in Problem 02 you will only work with model 3 and model 6 in this problem.

3a)

You will perform the Bayesian analysis using the Laplace Approximation just as you did in the previous assignment. You will define the log-posterior function just as you did in the previous assignment and so before doing so you must create the list of required information. This list will include the observed response, the design matrix, and the prior specification. You will use independent Gaussian priors on the regression parameters with a shared prior mean and shared prior standard deviation. You will use an Exponential prior on the unknown likelihood noise (the \(\sigma\) parameter).

Complete the two code chunks below. In the first, create the design matrix following mod03’s formula, and assign the object to the X03 variable. Complete the info_03_weak list by assigning the response to yobs and the design matrix to design_matrix. Specify the shared prior mean, mu_beta, to be 0, the shared prior standard deviation, tau_beta, as 50, and the rate parameter on the noise, sigma_rate, to be 1.

Complete the second code chunk with the same prior specification. The second code chunk however requires that you create the design matrix associated with mod06’s formula and assign the object to the X06 variable. Assign X06 to the design_matrix field of the info_06_weak list.

SOLUTION

X03 <- model.matrix(y ~ (x1 + I(x1^2))*(x2 + I(x2^2)), data = df)

info_03_weak <- list(
  yobs = df$y,
  design_matrix = X03,
  mu_beta = 0,
  tau_beta = 50,
  sigma_rate = 1
)

X06 <- model.matrix(y ~ splines::ns(x1, df = 12) * (x2 +I(x2^2) + I(x2^3) + I(x2^4)), data = df)

info_06_weak <- list(
  yobs = df$y,
  design_matrix = X06,
  mu_beta = 0,
  tau_beta = 50,
  sigma_rate = 1
)

3b)

You will now define the log-posterior function lm_logpost(). You will continue to use the log-transformation on \(\sigma\), and so you will actually define the log-posterior in terms of the mean trend \(\boldsymbol{\beta}\)-parameters and the unbounded noise parameter, \(\varphi = \log\left[\sigma\right]\).

The comments in the code chunk below tell you what you need to fill in. The unknown parameters to learn are contained within the first input argument, unknowns. You will assume that the unknown \(\boldsymbol{\beta}\)-parameters are listed before the unknown \(\varphi\) parameter in the unknowns vector. You must specify the number of \(\boldsymbol{\beta}\) parameters programmatically to allow scaling up your function to an arbitrary number of unknowns. You will assume that all variables contained in the my_info list (the second argument to lm_logpost()) are the same fields in the info_03_weak list you defined in Problem 3a).

Define the log-posterior function by completing the code chunk below. You must calculate the mean trend, mu, using matrix math between the design matrix and the unknown \(\boldsymbol{\beta}\) column vector.

HINT: This function should look very famaliar…

SOLUTION

lm_logpost <- function(unknowns, my_info)
{
  length_beta <- length(unknowns) - 1
  
  # extract the beta parameters from the `unknowns` vector
  beta_v <- unknowns[1:length_beta] %>% as.matrix()
  
  # extract the unbounded noise parameter, varphi
  lik_varphi <- unknowns[length(unknowns)]
  
  # back-transform from varphi to sigma
  lik_sigma <- exp(lik_varphi)
  
  # extract design matrix
  X <- my_info$design_matrix
  
  # calculate the linear predictor
  mu <- X %*% beta_v
  
  # evaluate the log-likelihood
  log_lik <- sum(dnorm(my_info$yobs, mean = mu, sd = lik_sigma, log = TRUE))
  
  # evaluate the log-prior
  log_prior_beta <- sum(dnorm(beta_v, mean = my_info$mu_beta, sd = my_info$tau_beta, log = TRUE))
  
  log_prior_sigma <- dexp(lik_sigma, rate = my_info$sigma_rate, log = TRUE)
  
  # add the mean trend prior and noise prior together
  log_prior <- log_prior_beta + log_prior_sigma
  
  # account for the transformation
  log_derive_adjust <- lik_varphi
  
  # sum together
  log_lik + log_prior + lik_varphi
  
}

3c)

The my_laplace() function is defined for you in the code chunk below. This function executes the laplace approximation and returns the object consisting of the posterior mode, posterior covariance matrix, and the log-evidence.

my_laplace <- function(start_guess, logpost_func, ...)
{
  # code adapted from the `LearnBayes`` function `laplace()`
  fit <- optim(start_guess,
               logpost_func,
               gr = NULL,
               ...,
               method = "BFGS",
               hessian = TRUE,
               control = list(fnscale = -1, maxit = 1001))
  
  mode <- fit$par
  post_var_matrix <- -solve(fit$hessian)
  p <- length(mode)
  int <- p/2 * log(2 * pi) + 0.5 * log(det(post_var_matrix)) + logpost_func(mode, ...)
  # package all of the results into a list
  list(mode = mode,
       var_matrix = post_var_matrix,
       log_evidence = int,
       converge = ifelse(fit$convergence == 0,
                         "YES", 
                         "NO"),
       iter_counts = as.numeric(fit$counts[1]))
}

Execute the Laplace Approximation for the model 3 formulation and the model 6 formulation. Assign the model 3 result to the laplace_03_weak object, and assign the model 6 result to the laplace_06_weak object. Check that the optimization scheme converged.

SOLUTION

start_guess_03 <- c(rnorm(info_03_weak$design_matrix %>% ncol()), 5)
start_guess_06 <- c(rnorm(info_06_weak$design_matrix %>% ncol()), 5)

### add more code chunks if you like
laplace_03_weak <- my_laplace(start_guess_03,
                              lm_logpost,
                              info_03_weak)
laplace_03_weak

## $mode
##  [1]  0.665292190  0.164471932 -0.160370454 -0.051975252 -0.556456896
##  [6]  0.122792393 -0.082521271  0.005338202  0.020383961 -0.398805777
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  1.382439e-02 -2.857431e-04 -6.974406e-03  3.995513e-04 -5.817154e-03
##  [2,] -2.857431e-04  7.785559e-03  1.697286e-04  8.041367e-04 -5.563797e-04
##  [3,] -6.974406e-03  1.697286e-04  7.581133e-03  1.620084e-04  3.406225e-03
##  [4,]  3.995513e-04  8.041367e-04  1.620084e-04  8.160218e-03  5.829452e-04
##  [5,] -5.817154e-03 -5.563797e-04  3.406225e-03  5.829452e-04  5.670717e-03
##  [6,]  2.435675e-03 -2.044357e-05  1.411762e-04  6.038636e-04 -1.933566e-03
##  [7,] -1.669154e-03 -2.966175e-03  6.048780e-04 -1.077101e-03  1.525058e-03
##  [8,]  1.328242e-03  1.841321e-04 -5.911626e-04 -2.460648e-03 -1.598428e-03
##  [9,]  2.527204e-03  3.147396e-04 -3.492292e-03 -9.078418e-04 -2.395753e-03
## [10,] -6.188075e-08 -8.510241e-09  5.193266e-08  4.787207e-08  5.758501e-08
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,]  2.435675e-03 -1.669154e-03  1.328242e-03  2.527204e-03 -6.188075e-08
##  [2,] -2.044357e-05 -2.966175e-03  1.841321e-04  3.147396e-04 -8.510241e-09
##  [3,]  1.411762e-04  6.048780e-04 -5.911626e-04 -3.492292e-03  5.193266e-08
##  [4,]  6.038636e-04 -1.077101e-03 -2.460648e-03 -9.078418e-04  4.787207e-08
##  [5,] -1.933566e-03  1.525058e-03 -1.598428e-03 -2.395753e-03  5.758501e-08
##  [6,]  1.216969e-02 -4.522204e-03  4.252484e-03 -2.772279e-03  1.666019e-08
##  [7,] -4.522204e-03  4.978960e-03 -2.548154e-03  8.311601e-04 -6.159891e-09
##  [8,]  4.252484e-03 -2.548154e-03  4.465035e-03 -4.413527e-04 -2.726561e-08
##  [9,] -2.772279e-03  8.311601e-04 -4.413527e-04  3.461300e-03 -3.111312e-08
## [10,]  1.666019e-08 -6.159891e-09 -2.726561e-08 -3.111312e-08  4.999649e-03
## 
## $log_evidence
## [1] -164.7934
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 123

laplace_06_weak <- my_laplace(start_guess_06,
                              lm_logpost,
                              info_06_weak)
laplace_06_weak

## $mode
##  [1]  -6.6484595   3.4962633   9.0989632   4.5605887   9.0125568   6.8649271
##  [7]   7.6500717   7.3278346   7.5069870   4.8386197   4.8822841  16.7361496
## [13]   2.3932857   1.4046485   7.3245263  -1.5132287  -2.8215250  -0.4551382
## [19]  -3.9582350   1.6387343  -2.7248122   0.2896645  -1.8434055  -1.9327134
## [25]  -3.5033200  -0.1594400  -5.7746471   3.5040164  12.9820301   1.5039461
## [31] -16.5476990   3.6503851 -12.5042022  -4.8727732  -9.8597094 -10.4641293
## [37]  -7.4626111  10.8662320 -11.8003938 -21.7176819   0.9385730  -1.1289199
## [43]   3.1357242  -2.0794339   2.9141041  -0.1445471   2.4543364   2.7035451
## [49]  -4.3120943  17.7212086  -4.8113605 -22.8717162 -39.0220452   0.3723432
## [55]   4.9885424   0.1148528   3.6193377   2.4902689   2.2639822   7.1160764
## [61]  -7.4197172  -7.1080440   4.3518312  21.8012963  23.0560070  -0.7575425
## 
## $var_matrix
##                [,1]          [,2]         [,3]          [,4]          [,5]
##  [1,]  13.955694350 -11.102762573 -15.43148649 -12.797002304 -14.648632092
##  [2,] -11.102762573  10.194689588  11.51446999  10.733707353  11.348153488
##  [3,] -15.431486493  11.514469987  17.66663342  13.678032610  16.454327361
##  [4,] -12.797002304  10.733707353  13.67803261  12.550161967  12.990218706
##  [5,] -14.648632092  11.348153488  16.45432736  12.990218706  15.825030183
##  [6,] -13.589283709  10.994001775  14.86643111  12.792036757  13.921181271
##  [7,] -14.172633004  11.149071909  15.78293221  12.764366104  15.099903083
##  [8,] -13.866391483  11.092946474  15.27971403  12.823468623  14.445503330
##  [9,] -14.034541892  11.134143952  15.54644299  12.812939782  14.785920834
## [10,] -13.778910688  10.998562656  15.20639531  12.675200993  14.428941274
## [11,]  -7.314751552   6.219533640   7.86703853   6.853745311   7.608102257
## [12,] -30.182786102  23.004292381  33.94082552  27.281567678  31.888884233
## [13,]  -3.976483671   3.790625345   4.04423277   3.903597866   4.011779850
## [14,] -17.616397897  17.059021874  17.69271231  16.900366488  18.078816545
## [15,]  -5.451030165   1.273024871   7.68713039   4.205107484   6.157389308
## [16,]   9.909650357  -9.848645254  -9.77248337  -9.635941920 -10.106471751
## [17,]  -0.759044598   2.008961109   0.11194327   1.013471878   0.617044419
## [18,]  17.063250805 -16.643762681 -17.06211592 -16.480372164 -17.475822229
## [19,]  18.063796219 -17.166866963 -18.43731827 -16.896244906 -18.672580838
## [20,]  16.901093565 -16.938914984 -16.45365961 -17.289458026 -16.941577154
## [21,]  17.804119225 -17.047929934 -18.06328437 -16.691138698 -18.419997152
## [22,]  17.514884801 -17.124664139 -17.42418007 -17.235694535 -17.704477340
## [23,]  18.024551420 -17.135342703 -18.36948310 -16.834072800 -18.819740993
## [24,]  17.612128673 -17.048486171 -17.69045433 -16.893588006 -18.118147294
## [25,]  17.581559500 -17.067477476 -17.62464316 -16.927333537 -17.961785612
## [26,]  17.662533040 -17.036260844 -17.78288156 -16.911162072 -18.178700298
## [27,]  10.553336000  -9.978989839 -10.74470517 -10.028054015 -10.864855703
## [28,]  34.933780037 -34.394389140 -34.75630821 -33.704511776 -35.746009554
## [29,]   5.639526332  -5.499723543  -5.57407574  -5.585278488  -5.742769766
## [30,]   0.063331179  -0.849283800   0.36538347  -0.994651535   0.438597799
## [31,]   8.040312116  -1.016040777 -12.48538643  -4.594179368  -9.810307699
## [32,]   2.168369387  -0.647248240  -2.29934532  -4.355221411  -1.274519786
## [33,]   7.209262498  -1.768795725 -10.41560810  -4.429895079  -9.123874148
## [34,]   4.936019159  -1.280269144  -6.71212547  -4.761315593  -4.737023137
## [35,]   5.568448278  -1.163908466  -8.02975996  -3.680583456  -6.810882439
## [36,]   5.363801199  -1.393234133  -7.41545692  -4.530640311  -5.711911985
## [37,]   5.547885005  -1.243484364  -7.90240172  -4.022686727  -6.485801071
## [38,]   4.369824271  -0.585393445  -6.32934695  -3.452734126  -4.896588180
## [39,]   1.870256569  -0.970715789  -2.37628965  -1.710420735  -1.968293538
## [40,]  14.411171048  -2.091994209 -21.00923451 -10.301396199 -16.687025846
## [41,]  -2.866588883   1.495017343   3.62797769   1.988858723   3.407166161
## [42,]  -7.615807691   8.629924812   6.98914328   7.822602615   7.535635558
## [43,] -10.814637403  10.351901788  10.87800005  10.229894577  11.154313171
## [44,]  -8.753051725   9.510321748   7.98796127   9.483323036   8.497264834
## [45,] -10.432200160  10.004887147  10.57327291   9.739540568  10.862393981
## [46,]  -9.703408430   9.853845996   9.38249886   9.841150835   9.593547248
## [47,] -10.344544250   9.945948486  10.47058831   9.631372289  10.878646662
## [48,] -10.019356690   9.858608379   9.96507361   9.583136248  10.373082811
## [49,]  -9.388086067   9.594583538   9.03957676   9.479034348   9.289797789
## [50,] -10.733366051  10.301237754  10.85678663  10.169844010  11.119702623
## [51,]  -4.468201765   4.385318660   4.41655054   4.380063461   4.516162380
## [52,] -19.665507969  20.761224845  18.58945994  19.760807857  19.727293258
## [53,]   4.360412067  -2.972063169  -5.30274767  -3.287060437  -4.903149591
## [54,]   1.534652822  -1.452967358  -1.59693282  -1.225519804  -1.701071509
## [55,]   0.360873971  -2.305200574   0.83290584  -1.198733960   0.049516742
## [56,]   1.417049892  -2.046856206  -1.28501216  -0.690686668  -1.682978780
## [57,]   0.470552673  -1.965057063   0.37743091  -1.120404069  -0.039781553
## [58,]   0.496015366  -1.812761275   0.15829432  -0.502215393  -0.576729193
## [59,]   1.160449771  -2.238805588  -0.60671650  -1.487156753  -0.941314020
## [60,]   0.545447340  -1.775160767   0.06599072  -0.653187576  -0.557448131
## [61,]   1.398485892  -2.519315506  -0.82006357  -1.664448300  -1.178312474
## [62,]   1.210464137  -2.254010473  -0.70983594  -1.298327996  -1.172483669
## [63,]   0.319353424  -0.419379264  -0.27141649  -0.256187050  -0.340418547
## [64,]   0.297834316  -4.852509384   2.15412232  -1.726512965   0.473911765
## [65,]  -2.809501119   2.523497171   3.03638984   2.492064039   2.953538562
## [66,]   0.008110337  -0.003305448  -0.01084178  -0.006145322  -0.009196886
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,] -13.589283709 -14.172633004 -13.866391483 -14.034541892 -13.778910688
##  [2,]  10.994001774  11.149071909  11.092946474  11.134143952  10.998562656
##  [3,]  14.866431108  15.782932208  15.279714029  15.546442995  15.206395314
##  [4,]  12.792036757  12.764366104  12.823468623  12.812939781  12.675200993
##  [5,]  13.921181271  15.099903083  14.445503330  14.785920834  14.428941274
##  [6,]  13.778726255  13.405693134  13.725913464  13.552716596  13.478330072
##  [7,]  13.405693134  15.295689075  13.402286447  14.608360142  13.806778925
##  [8,]  13.725913464  13.402286447  14.649169506  13.473326522  13.925288215
##  [9,]  13.552716596  14.608360142  13.473326522  14.742673349  13.395484526
## [10,]  13.478330072  13.806778925  13.925288215  13.395484526  14.897966476
## [11,]   7.137587705   7.516957025   7.117039760   7.691306627   6.371284356
## [12,]  29.282913514  30.653990786  30.071761104  30.131491829  30.262614703
## [13,]   4.026542496   3.735043649   4.318836136   3.326129647   5.374069739
## [14,]  17.505622463  17.642375355  17.672698780  17.672774763  17.379336876
## [15,]   4.955870211   5.782115558   5.257458613   5.531680579   5.356681646
## [16,]  -9.888054746  -9.891012062  -9.958057736  -9.934023475  -9.780318634
## [17,]   0.891925828   0.667416689   0.824074306   0.742758944   0.760100594
## [18,] -16.982335654 -17.063658187 -17.124050400 -17.109110742 -16.848822668
## [19,] -17.833046655 -18.210923749 -18.074960244 -18.153569344 -17.789153516
## [20,] -17.114113509 -16.639069601 -17.072677911 -16.882265425 -16.740304119
## [21,] -17.551289634 -17.974806134 -17.808380128 -17.897269656 -17.535107377
## [22,] -17.696768526 -17.273530034 -17.606715267 -17.571817639 -17.192614237
## [23,] -17.606986108 -18.320225314 -18.055015421 -18.079122645 -17.903809722
## [24,] -17.287769615 -18.621093489 -16.337528751 -18.463671759 -16.930511477
## [25,] -17.749632990 -16.440267714 -19.165673209 -16.643956626 -18.183921122
## [26,] -17.358853289 -18.540089693 -16.624406000 -18.492542984 -15.210581901
## [27,] -10.561766160 -10.043919600 -11.320556976 -10.258292440 -10.974857848
## [28,] -34.746719211 -35.170410512 -34.725789575 -35.038393514 -35.220983537
## [29,]  -5.495845657  -6.404657008  -4.540004353  -5.886168091  -6.465867251
## [30,]  -0.250863774   0.083526396  -0.089328554  -0.013691932  -0.172094830
## [31,]  -6.874942926  -8.908385584  -7.612960427  -8.261909811  -7.767534510
## [32,]  -2.945786647  -1.502361698  -2.412521391  -2.001258250  -2.280792874
## [33,]  -5.758430700  -8.248085130  -6.698518738  -7.462336834  -6.981206389
## [34,]  -5.726199893  -4.296205297  -5.189839283  -4.796122862  -4.915678870
## [35,]  -4.200279876  -7.356390073  -4.342594520  -6.194156578  -5.253330717
## [36,]  -5.523287555  -3.934970218  -7.459714937  -4.209337933  -5.341304760
## [37,]  -4.584732968  -7.309351758  -3.424850376  -7.994425910  -4.887011101
## [38,]  -4.007984739  -4.595929968  -4.189884167  -3.469931997  -9.068748510
## [39,]  -1.760146190  -1.876732407  -1.892018531  -2.498395386   0.749059955
## [40,] -12.984627017 -15.179120211 -14.213646146 -13.612986974 -16.317072493
## [41,]   2.478268090   3.437797033   2.236620921   5.149361088  -2.870237163
## [42,]   7.730391474   7.508970037   7.697038372   7.611090688   7.550164276
## [43,]  10.703982481  10.866042958  10.837506854  10.858060475  10.656566556
## [44,]   8.993992747   8.514541711   8.884890270   8.722012101   8.703365255
## [45,]  10.273448475  10.529730931  10.444506913  10.479053349  10.273137925
## [46,]   9.975479491   9.462672273   9.745682867   9.771058045   9.439527686
## [47,]  10.047277005  10.451919076  10.507105593  10.257592879  10.410032593
## [48,]   9.586288837  11.542695841   8.052693230  11.304078581   9.200818213
## [49,]  10.043608641   6.916962059  12.540589659   6.966767639  11.459951263
## [50,]  10.382240371  11.883464118   9.445969320  11.899485740   5.276705317
## [51,]   4.640521011   3.583008387   5.567259481   4.000819995   6.731023521
## [52,]  19.785555383  20.019620271  18.976200988  18.445126711  25.637822411
## [53,]  -4.273745388  -3.022066490  -6.516671679  -6.189028939   4.677256558
## [54,]  -1.483784665  -1.573599304  -1.537114888  -1.551956238  -1.488946192
## [55,]  -0.682559600  -0.116306016  -0.492904747  -0.306699256  -0.401046006
## [56,]  -1.149441519  -1.643200222  -1.343596064  -1.473662296  -1.384448510
## [57,]  -0.880942795  -0.169387820  -0.625413849  -0.409998589  -0.485679736
## [58,]  -0.210755559  -0.718425375  -0.454553784  -0.508029383  -0.575753863
## [59,]  -1.551482679  -0.633783644  -1.543308009  -1.002787373  -1.081416446
## [60,]  -0.307234976  -1.650388296   0.969868665  -1.303141834  -1.012607588
## [61,]  -1.898448023   0.203687910  -3.523129227   0.006643482  -0.037201541
## [62,]  -1.184068294  -1.635682417  -0.697188393  -1.823848584   1.789023374
## [63,]  -0.402094188   0.066440677  -0.812262602  -0.030083392  -1.613234053
## [64,]  -0.815717359  -0.249792986   0.010966215   0.185375875  -3.184982183
## [65,]   2.861113200   1.998142064   4.057942074   3.300665820  -1.115185762
## [66,]  -0.007456047  -0.008228841  -0.008471336  -0.008762322  -0.004737211
##               [,11]         [,12]         [,13]         [,14]        [,15]
##  [1,]  -7.314751552 -30.182786102  -3.976483671  -17.61639790  -5.45103016
##  [2,]   6.219533640  23.004292381   3.790625345   17.05902187   1.27302487
##  [3,]   7.867038528  33.940825522   4.044232772   17.69271231   7.68713039
##  [4,]   6.853745311  27.281567678   3.903597866   16.90036649   4.20510748
##  [5,]   7.608102257  31.888884233   4.011779850   18.07881655   6.15738931
##  [6,]   7.137587705  29.282913514   4.026542496   17.50562246   4.95587021
##  [7,]   7.516957025  30.653990786   3.735043649   17.64237536   5.78211556
##  [8,]   7.117039760  30.071761103   4.318836136   17.67269878   5.25745861
##  [9,]   7.691306627  30.131491829   3.326129647   17.67277476   5.53168058
## [10,]   6.371284356  30.262614703   5.374069739   17.37933688   5.35668165
## [11,]   5.122348579  14.397227060  -0.100430496   10.56028996   1.70241370
## [12,]  14.397227060  67.956607566  11.316372592   34.81098856  14.68941622
## [13,]  -0.100430496  11.316372592   9.032164607    7.11619584  -0.44917553
## [14,]  10.560289957  34.810988562   7.116195839   64.88544183 -27.78832547
## [15,]   1.702413705  14.689416219  -0.449175533  -27.78832547  37.96342213
## [16,]  -6.105559245 -19.176780490  -4.221092531  -38.39096124  16.25089317
## [17,]   0.873219523   0.440160170   1.093447829   17.40002246 -17.83892981
## [18,] -10.238844393 -33.686023935  -6.922325178  -59.79608648  25.85650706
## [19,] -10.732282614 -35.948239903  -7.182802700  -66.88872183  28.59772455
## [20,] -10.308076618 -32.920240386  -7.058127428  -63.36275111  27.73185487
## [21,] -10.606232687 -35.358215506  -7.144674066  -65.33978619  27.83233627
## [22,] -10.602503529 -34.444063600  -7.064493247  -64.81823246  27.93836403
## [23,] -10.620618660 -35.935654272  -7.282198717  -65.26772966  27.43326927
## [24,] -10.750231021 -34.741343508  -6.858484681  -64.72683741  27.67222708
## [25,] -10.177633755 -34.860281103  -7.651370085  -65.03533956  27.96547269
## [26,] -11.779562041 -34.443070660  -5.359203888  -64.52590106  27.39241645
## [27,]  -5.082298417 -21.885361355  -6.257686386  -35.91491407  14.36018360
## [28,] -21.160602082 -68.419401800 -14.450204818 -136.18233156  60.97030035
## [29,]  -4.140886857 -10.590480771  -0.670195021  -17.86336368   6.79708489
## [30,]  -0.071550570  -0.008893207   0.070960801   25.63857574 -24.56884549
## [31,]  -2.342908735 -22.155222848   0.809296952   30.68840031 -44.71399218
## [32,]  -0.539880930  -6.089879560   0.514005135   29.86313915 -34.27968827
## [33,]  -2.398100042 -19.147467973   0.365844623   26.38778197 -39.61549997
## [34,]  -1.523092392 -13.267353479   0.343246230   27.57431010 -36.89427073
## [35,]  -1.848160897 -15.009483861   0.793646298   28.20890649 -38.55736032
## [36,]  -1.546268600 -14.541005208  -0.068956640   27.30478621 -37.40114172
## [37,]  -2.205410910 -14.723181057   1.249348055   27.76702953 -38.08132452
## [38,]   1.007309588 -12.184865848  -0.699667119   30.19196812 -38.27289129
## [39,]  -3.405769823  -2.817184067   4.043473049   13.11874520 -16.71592656
## [40,]   0.408142712 -47.881541695 -13.308692175   62.21155248 -88.83755021
## [41,]  10.066203002  -6.346666208 -28.992905909   12.54367224  -7.00023096
## [42,]   5.015075599  13.919880756   3.774830840   31.87525723 -16.09256761
## [43,]   6.541578026  21.237982139   4.420183182   41.09383245 -16.67873866
## [44,]   5.654001680  16.251084199   4.118671649   36.76761580 -16.83483046
## [45,]   6.306757367  20.499197510   4.275797517   39.08887058 -15.97051010
## [46,]   6.098167891  18.596709316   4.198687096   38.36495846 -16.60500867
## [47,]   6.185099766  20.309885403   4.305996306   38.69054035 -15.77919754
## [48,]   6.352810320  19.504522673   4.122648776   38.38894701 -16.08901572
## [49,]   5.235514018  17.761316873   4.274401108   37.44696957 -16.25256101
## [50,]   8.439999234  21.575719262   3.838927752   39.83309290 -16.16121305
## [51,]   1.402010383   8.181139984   1.915534073   15.62916296  -5.47269520
## [52,]   7.406291868  44.272948791  25.164608044   86.56750299 -42.06740700
## [53,] -10.007632607   3.109082594  26.257889124   -9.95251249   1.87053873
## [54,]  -0.896359329  -3.088446802  -0.687236204  -14.34257481  12.49679739
## [55,]  -0.862980669   0.933863337  -1.285459660  -18.96774971  20.02330368
## [56,]  -1.073527961  -2.239071656  -1.067698345  -17.67153189  16.90781355
## [57,]  -0.773773415   0.323022899  -1.097372415  -17.25735414  18.20932520
## [58,]  -0.715112631   0.128658989  -1.011493769  -16.71217776  17.62995144
## [59,]  -1.061617513  -1.392523406  -1.240276869  -18.16193759  17.87662432
## [60,]  -0.650712135   0.057070805  -0.885423345  -16.54834486  17.45573754
## [61,]  -1.502510999  -1.986500276  -1.429042626  -19.16642493  18.29908490
## [62,]  -2.263676744  -1.819440907  -0.910891008  -18.21573252  17.81321303
## [63,]   0.781671838  -0.614716155  -0.714374360   -6.14711475   6.40145985
## [64,]  -0.107490091   1.583452527  -7.538866852  -41.45782536  45.30092801
## [65,]   3.957504439   2.109502805  -6.262101788    6.24466357  -0.11857121
## [66,]  -0.004955396  -0.018329263   0.004656771    0.01777169  -0.03141706
##              [,16]        [,17]        [,18]        [,19]        [,20]
##  [1,]   9.90965036  -0.75904460  17.06325081  18.06379622  16.90109357
##  [2,]  -9.84864525   2.00896111 -16.64376268 -17.16686696 -16.93891498
##  [3,]  -9.77248337   0.11194327 -17.06211592 -18.43731827 -16.45365961
##  [4,]  -9.63594192   1.01347188 -16.48037216 -16.89624491 -17.28945803
##  [5,] -10.10647175   0.61704442 -17.47582223 -18.67258084 -16.94157715
##  [6,]  -9.88805475   0.89192583 -16.98233565 -17.83304666 -17.11411351
##  [7,]  -9.89101206   0.66741669 -17.06365819 -18.21092375 -16.63906960
##  [8,]  -9.95805774   0.82407431 -17.12405040 -18.07496024 -17.07267791
##  [9,]  -9.93402348   0.74275894 -17.10911074 -18.15356934 -16.88226543
## [10,]  -9.78031863   0.76010059 -16.84882267 -17.78915352 -16.74030412
## [11,]  -6.10555925   0.87321952 -10.23884439 -10.73228261 -10.30807662
## [12,] -19.17678049   0.44016017 -33.68602394 -35.94823990 -32.92024039
## [13,]  -4.22109253   1.09344783  -6.92232518  -7.18280270  -7.05812743
## [14,] -38.39096124  17.40002246 -59.79608648 -66.88872183 -63.36275111
## [15,]  16.25089317 -17.83892981  25.85650706  28.59772455  27.73185487
## [16,]  23.20033881  -9.70598228  35.22891159  39.43209498  37.70930839
## [17,]  -9.70598228   9.53529323 -16.51593003 -17.98882892 -17.06398574
## [18,]  35.22891159 -16.51593003  56.56350359  60.92875793  58.89333779
## [19,]  39.43209498 -17.98882892  60.92875793  70.60070465  63.57107001
## [20,]  37.70930839 -17.06398574  58.89333779  63.57107001  65.39011088
## [21,]  38.57091482 -17.52461487  60.03144176  68.05666073  62.40389187
## [22,]  38.42234884 -17.40242058  59.88863123  66.21641277  64.50725902
## [23,]  38.52425647 -17.37962502  60.04551009  67.76251396  62.73217060
## [24,]  38.30366281 -17.33790308  59.66942200  66.64096942  63.35574957
## [25,]  38.48531802 -17.47508861  59.93333989  67.04999164  63.51606153
## [26,]  38.17248044 -17.21930979  59.47431879  66.48426472  63.05549674
## [27,]  21.19364136  -9.35159126  33.41905652  36.99681064  34.88603274
## [28,]  80.68782614 -37.28040624 124.69068906 140.61228637 133.14418414
## [29,]  10.60156571  -4.52852237  17.02628731  17.80070667  18.29561067
## [30,] -14.40156338  12.69822820 -23.03513774 -27.43960684 -23.62039366
## [31,] -18.72662620  20.05355081 -28.97442277 -29.07335596 -34.36158143
## [32,] -16.77671972  17.23671739 -27.28601735 -33.68677364 -23.70017637
## [33,] -15.71657347  18.05997765 -24.73914249 -26.11629309 -28.63150415
## [34,] -15.98744437  17.52045679 -25.48086343 -29.18134488 -25.86729629
## [35,] -16.57080110  18.05638711 -26.37487125 -28.48726048 -29.22860803
## [36,] -15.90572823  17.62405756 -25.34003212 -28.41329891 -26.60349363
## [37,] -16.28379172  17.84223764 -25.89558238 -28.31897840 -28.22964345
## [38,] -17.57866466  18.29073143 -28.00409936 -31.45926792 -29.31316417
## [39,]  -7.19174036   8.32094039 -12.03786027 -13.64775736 -12.84879106
## [40,] -37.63523575  40.46558578 -58.37672059 -63.26255210 -63.45843638
## [41,]  -6.62406177   4.65921773 -11.32771499 -13.45880768 -11.30368321
## [42,] -19.22635114   9.30263171 -30.16595715 -32.39333937 -31.74503306
## [43,] -24.79029628  10.09741696 -37.38269946 -42.67804153 -39.69298918
## [44,] -22.45226409   9.59491901 -33.98407998 -36.67364926 -38.43582481
## [45,] -23.51869376   9.74769463 -35.78072845 -40.58390151 -37.45761386
## [46,] -23.24220447   9.79623176 -35.29346920 -39.02507154 -38.48246504
## [47,] -23.29841720   9.62394471 -35.44405526 -40.07107984 -37.27974635
## [48,] -23.17307342   9.66966434 -35.20351638 -39.53275706 -37.48001203
## [49,] -22.71517458   9.54053463 -34.43156148 -38.12755587 -37.47892220
## [50,] -23.96717899   9.90160416 -36.50779930 -41.20960870 -38.51074098
## [51,]  -9.56035523   3.37051174 -14.43444105 -15.93542142 -15.48966654
## [52,] -52.31899819  24.04020200 -79.32690609 -88.38591217 -86.37827752
## [53,]   5.45055081  -2.19853094   8.86454753  11.54854651   7.26872745
## [54,]   7.83051162  -7.05505959  13.23792389  15.14105989  13.52670452
## [55,]  10.80696903 -10.40701698  18.13116721  18.92320997  19.67238041
## [56,]   9.68979579  -9.34291949  16.58203747  19.21750231  15.40656316
## [57,]   9.68407299  -9.61126523  16.45239638  17.52037325  17.59970623
## [58,]   9.30385038  -9.37014650  15.80886995  17.51511428  15.95035160
## [59,]  10.12770695  -9.66296320  17.26686240  18.66062294  17.99300722
## [60,]   9.19086733  -9.30962349  15.70608710  17.19709034  16.04424628
## [61,]  10.73075572  -9.91189735  18.14378202  19.86527842  18.69339149
## [62,]  10.13813401  -9.65634933  17.23654066  19.00826577  17.49403634
## [63,]   3.19208780  -3.60310090   5.78056256   6.45054139   5.87038487
## [64,]  23.69848309 -23.38242511  39.64783805  42.20673102  41.93658395
## [65,]  -3.96736832   0.43674108  -5.68389550  -6.89088737  -5.13773343
## [66,]  -0.01086202   0.01357292  -0.01648165  -0.01747131  -0.01945263
##              [,21]        [,22]        [,23]       [,24]       [,25]
##  [1,]  17.80411923  17.51488480  18.02455142  17.6121287  17.5815595
##  [2,] -17.04792993 -17.12466414 -17.13534270 -17.0484862 -17.0674775
##  [3,] -18.06328437 -17.42418007 -18.36948310 -17.6904543 -17.6246432
##  [4,] -16.69113870 -17.23569454 -16.83407280 -16.8935880 -16.9273335
##  [5,] -18.41999715 -17.70447734 -18.81974099 -18.1181473 -17.9617856
##  [6,] -17.55128963 -17.69676853 -17.60698611 -17.2877696 -17.7496330
##  [7,] -17.97480613 -17.27353003 -18.32022531 -18.6210935 -16.4402677
##  [8,] -17.80838013 -17.60671527 -18.05501542 -16.3375288 -19.1656732
##  [9,] -17.89726966 -17.57181764 -18.07912265 -18.4636718 -16.6439566
## [10,] -17.53510738 -17.19261424 -17.90380972 -16.9305115 -18.1839211
## [11,] -10.60623269 -10.60250353 -10.62061866 -10.7502310 -10.1776338
## [12,] -35.35821551 -34.44406360 -35.93565427 -34.7413435 -34.8602811
## [13,]  -7.14467407  -7.06449325  -7.28219872  -6.8584847  -7.6513701
## [14,] -65.33978619 -64.81823246 -65.26772966 -64.7268374 -65.0353396
## [15,]  27.83233627  27.93836403  27.43326927  27.6722271  27.9654727
## [16,]  38.57091482  38.42234884  38.52425647  38.3036628  38.4853180
## [17,] -17.52461487 -17.40242058 -17.37962502 -17.3379031 -17.4750886
## [18,]  60.03144176  59.88863123  60.04551009  59.6694220  59.9333399
## [19,]  68.05666073  66.21641277  67.76251396  66.6409694  67.0499916
## [20,]  62.40389187  64.50725902  62.73217060  63.3557496  63.5160615
## [21,]  66.79455683  64.34847261  66.39410581  65.0601877  65.4977471
## [22,]  64.34847261  66.98650129  63.56590053  65.1372554  64.6329766
## [23,]  66.39410581  63.56590053  67.66258424  64.5568672  65.7334364
## [24,]  65.06018772  65.13725543  64.55686717  68.2745818  60.3942434
## [25,]  65.49774712  64.63297665  65.73343637  60.3942434  71.8257773
## [26,]  64.92840096  64.88497041  64.43197806  67.6914974  59.0499666
## [27,]  36.30626939  35.69327073  36.37066193  33.8827948  39.0805300
## [28,] 137.00239478 135.97480549 136.97967261 136.2474667 136.1238094
## [29,]  17.49042339  18.00657771  17.84776410  19.9005857  15.0677905
## [30,] -26.42807738 -25.18554835 -26.35002796 -25.4936224 -25.7217554
## [31,] -29.19366724 -32.18140545 -28.81437604 -30.7256543 -30.9157709
## [32,] -32.46084384 -27.37637613 -31.98136158 -29.5203935 -29.9345888
## [33,] -25.35848072 -27.99883241 -24.53924612 -26.3651706 -26.6556490
## [34,] -28.50451345 -25.67681488 -28.91074208 -27.3965044 -27.6077532
## [35,] -27.63439808 -29.80297265 -26.90714667 -26.9812179 -29.8270358
## [36,] -27.72327689 -26.85884213 -27.59951717 -30.9184762 -22.8896512
## [37,] -27.48905549 -28.19716819 -26.99190353 -23.1852037 -33.0966689
## [38,] -30.69082913 -30.60403443 -29.49640137 -31.2746904 -27.7870174
## [39,] -13.18688225 -12.90276413 -13.45792765 -12.8948131 -13.7436049
## [40,] -61.74005619 -63.12235510 -60.41379247 -61.8673445 -62.4748166
## [41,] -12.98610753 -12.64981251 -12.30779497 -12.3998668 -11.8572360
## [42,] -31.86951647 -32.01429820 -31.77494475 -31.8104871 -31.9687667
## [43,] -41.54632962 -40.89642288 -41.46049716 -40.9695178 -41.1968910
## [44,] -36.02529432 -37.54946305 -36.11209149 -36.8031784 -36.8374931
## [45,] -39.78908033 -38.60097959 -39.72047262 -38.8978243 -39.2337270
## [46,] -38.02150693 -40.15574716 -37.12483046 -38.7998351 -37.9519289
## [47,] -39.27716597 -37.39736016 -40.28778121 -37.7967610 -39.6311828
## [48,] -38.70844746 -38.61173517 -38.46148227 -43.7131281 -31.4861826
## [49,] -37.16614914 -37.29451418 -37.40850313 -28.0071489 -50.6646979
## [50,] -40.38023453 -40.50147880 -39.39120254 -45.2156620 -30.4388529
## [51,] -15.61994659 -15.30575203 -15.94175096 -11.7078100 -22.0239356
## [52,] -86.19497617 -86.39583259 -87.21111447 -87.4410801 -86.4858550
## [53,]  11.39659734  10.04695025   9.58893104   5.3456297  14.9755756
## [54,]  14.65393054  14.17588900  14.59849016  14.2776996  14.3872299
## [55,]  18.65829305  19.38298623  18.52605198  18.9499630  19.0550485
## [56,]  18.61100964  16.72761365  18.41562686  17.5286206  17.7257520
## [57,]  17.04448440  17.80598061  16.77641907  17.2459332  17.3329748
## [58,]  17.12445499  15.54270313  17.49941233  16.4507335  16.9525231
## [59,]  18.12732969  18.94339213  17.77344531  17.7845087  18.6374605
## [60,]  16.78650914  16.26907692  16.89212420  19.6582708  12.6994936
## [61,]  19.33285385  19.33704200  18.76034155  14.3113372  25.0088665
## [62,]  18.56112659  18.51877724  17.91389070  20.6801375  13.8692889
## [63,]   6.24342467   5.92463332   6.37432723   4.3990471   8.9800910
## [64,]  41.10686740  41.53758042  41.26232123  41.7942708  41.5625989
## [65,]  -6.90194958  -6.36586266  -6.01770489  -4.2244820  -8.4007981
## [66,]  -0.01696038  -0.01812897  -0.01724008  -0.0182767  -0.0177667
##              [,26]         [,27]         [,28]         [,29]         [,30]
##  [1,]  17.66253304  1.055334e+01   34.93378004    5.63952633   0.063331178
##  [2,] -17.03626084 -9.978990e+00  -34.39438914   -5.49972354  -0.849283800
##  [3,] -17.78288156 -1.074471e+01  -34.75630821   -5.57407574   0.365383473
##  [4,] -16.91116207 -1.002805e+01  -33.70451178   -5.58527849  -0.994651534
##  [5,] -18.17870030 -1.086486e+01  -35.74600956   -5.74276977   0.438597800
##  [6,] -17.35885329 -1.056177e+01  -34.74671921   -5.49584566  -0.250863774
##  [7,] -18.54008969 -1.004392e+01  -35.17041051   -6.40465701   0.083526396
##  [8,] -16.62440600 -1.132056e+01  -34.72578958   -4.54000435  -0.089328554
##  [9,] -18.49254298 -1.025829e+01  -35.03839351   -5.88616809  -0.013691932
## [10,] -15.21058190 -1.097486e+01  -35.22098354   -6.46586725  -0.172094830
## [11,] -11.77956204 -5.082298e+00  -21.16060208   -4.14088686  -0.071550570
## [12,] -34.44307066 -2.188536e+01  -68.41940180  -10.59048077  -0.008893206
## [13,]  -5.35920389 -6.257686e+00  -14.45020482   -0.67019502   0.070960801
## [14,] -64.52590106 -3.591491e+01 -136.18233156  -17.86336368  25.638575740
## [15,]  27.39241645  1.436018e+01   60.97030035    6.79708489 -24.568845492
## [16,]  38.17248044  2.119364e+01   80.68782614   10.60156571 -14.401563377
## [17,] -17.21930979 -9.351591e+00  -37.28040624   -4.52852237  12.698228205
## [18,]  59.47431879  3.341906e+01  124.69068906   17.02628731 -23.035137742
## [19,]  66.48426472  3.699681e+01  140.61228637   17.80070667 -27.439606845
## [20,]  63.05549674  3.488603e+01  133.14418414   18.29561067 -23.620393663
## [21,]  64.92840096  3.630627e+01  137.00239478   17.49042339 -26.428077380
## [22,]  64.88497041  3.569327e+01  135.97480549   18.00657771 -25.185548355
## [23,]  64.43197806  3.637066e+01  136.97967261   17.84776410 -26.350027959
## [24,]  67.69149736  3.388279e+01  136.24746674   19.90058570 -25.493622380
## [25,]  59.04996656  3.908053e+01  136.12380935   15.06779054 -25.721755353
## [26,]  74.73252033  2.963171e+01  136.05335595   23.19579908 -25.430833373
## [27,]  29.63171154  3.068008e+01   70.02782238   -5.65194317 -13.942064624
## [28,] 136.05335595  7.002782e+01  294.29358538   52.76401097 -54.722823591
## [29,]  23.19579908 -5.651943e+00   52.76401097   43.44433059  -5.886117260
## [30,] -25.43083337 -1.394206e+01  -54.72282359   -5.88611726  27.453213273
## [31,] -30.22543477 -1.533554e+01  -68.19705814   -8.64766886  22.513564450
## [32,] -29.42302209 -1.617493e+01  -64.22671273   -6.13491643  29.473021532
## [33,] -26.00254591 -1.322538e+01  -58.63200095   -7.09500032  22.403513662
## [34,] -27.09897707 -1.438787e+01  -60.35979206   -6.42459970  25.074546494
## [35,] -26.96222031 -1.514491e+01  -61.63428334   -6.21470579  24.455956681
## [36,] -29.36955246 -1.139080e+01  -62.05011642  -12.15445987  24.645420379
## [37,] -25.85391045 -1.592377e+01  -59.98776414   -4.11376722  24.302206694
## [38,] -37.14352122 -1.689959e+01  -57.54710678    7.63955531  26.039881097
## [39,] -10.03123080 -9.681124e+00  -31.54821910   -6.43055613  14.072872007
## [40,] -62.27703438 -3.172795e+01 -133.54244187   -8.93708545  48.695090966
## [41,] -16.32166860 -5.137283e+00  -17.39311684   10.39292738  11.220728776
## [42,] -31.65893308 -1.760383e+01  -66.94985073   -9.09126373  10.426735768
## [43,] -40.85472958 -2.271043e+01  -86.36651385  -11.14588328  16.177077226
## [44,] -36.57611685 -2.005046e+01  -77.63783462  -10.76320513  12.314223549
## [45,] -38.81802860 -2.172047e+01  -81.97262108  -10.49950410  15.312352693
## [46,] -38.68311871 -2.107174e+01  -80.40676313  -10.27975858  14.071796134
## [47,] -37.66296030 -2.158466e+01  -81.55065371  -11.06409398  14.983095944
## [48,] -43.14877445 -1.865876e+01  -80.75712676  -12.53087652  14.508929974
## [49,] -27.04863607 -2.355053e+01  -81.84942632  -13.62023038  13.650089160
## [50,] -58.53663642 -1.614624e+01  -77.41223776   -4.56820256  15.403424786
## [51,]  -2.61444631 -2.183355e+01  -30.27823495    8.80459724   4.939051255
## [52,] -84.62687826 -3.262130e+01 -211.50780394  -82.88175107  33.961548770
## [53,]   3.19968903  4.008214e+01  -29.83532631 -107.78781814  -5.695130849
## [54,]  14.22700876  7.842250e+00   30.45089517    3.57484164 -12.189586982
## [55,]  18.77600892  1.006777e+01   40.83407243    5.22571818 -12.575682727
## [56,]  17.43639154  9.638781e+00   37.66617487    4.34562349 -13.872805205
## [57,]  17.12332127  9.208407e+00   37.05119434    4.59322078 -12.299795490
## [58,]  16.26665129  8.932469e+00   36.10231331    4.64952499 -12.296899699
## [59,]  17.89450987  1.020898e+01   38.29241269    3.63475523 -13.243580031
## [60,]  18.15473300  5.722280e+00   38.68322521   11.91590690 -12.385852171
## [61,]  17.67400566  1.604992e+01   33.97534836  -10.84621480 -13.529636211
## [62,]  27.31402948  6.886313e+00   35.66709581    1.24035376 -13.288425822
## [63,]   0.05439311  1.002946e+01   11.61373668   -5.66482367  -5.841817358
## [64,]  40.19993084  1.456476e+01  103.36416120   40.08575353 -27.583993866
## [65,]  -3.30580031 -2.096014e+01   11.98167297   53.00660566   0.519300998
## [66,]  -0.01673708  5.567313e-04   -0.05578447   -0.03749309   0.016895861
##              [,31]        [,32]       [,33]        [,34]        [,35]
##  [1,]   8.04031212   2.16836939   7.2092625   4.93601916   5.56844828
##  [2,]  -1.01604078  -0.64724824  -1.7687957  -1.28026914  -1.16390847
##  [3,] -12.48538643  -2.29934532 -10.4156081  -6.71212547  -8.02975996
##  [4,]  -4.59417937  -4.35522141  -4.4298951  -4.76131559  -3.68058346
##  [5,]  -9.81030770  -1.27451979  -9.1238741  -4.73702314  -6.81088244
##  [6,]  -6.87494293  -2.94578665  -5.7584307  -5.72619989  -4.20027988
##  [7,]  -8.90838558  -1.50236170  -8.2480851  -4.29620530  -7.35639007
##  [8,]  -7.61296043  -2.41252139  -6.6985187  -5.18983928  -4.34259452
##  [9,]  -8.26190981  -2.00125825  -7.4623368  -4.79612286  -6.19415658
## [10,]  -7.76753451  -2.28079287  -6.9812064  -4.91567887  -5.25333072
## [11,]  -2.34290873  -0.53988093  -2.3981000  -1.52309239  -1.84816090
## [12,] -22.15522285  -6.08987956 -19.1474680 -13.26735348 -15.00948386
## [13,]   0.80929695   0.51400513   0.3658446   0.34324623   0.79364630
## [14,]  30.68840031  29.86313916  26.3877820  27.57431010  28.20890649
## [15,] -44.71399218 -34.27968827 -39.6155000 -36.89427073 -38.55736032
## [16,] -18.72662620 -16.77671972 -15.7165735 -15.98744437 -16.57080110
## [17,]  20.05355081  17.23671739  18.0599777  17.52045679  18.05638711
## [18,] -28.97442277 -27.28601735 -24.7391425 -25.48086343 -26.37487125
## [19,] -29.07335596 -33.68677365 -26.1162931 -29.18134488 -28.48726048
## [20,] -34.36158143 -23.70017637 -28.6315042 -25.86729629 -29.22860803
## [21,] -29.19366724 -32.46084385 -25.3584807 -28.50451345 -27.63439808
## [22,] -32.18140545 -27.37637613 -27.9988324 -25.67681488 -29.80297265
## [23,] -28.81437604 -31.98136158 -24.5392461 -28.91074208 -26.90714667
## [24,] -30.72565434 -29.52039346 -26.3651706 -27.39650442 -26.98121788
## [25,] -30.91577087 -29.93458877 -26.6556490 -27.60775319 -29.82703580
## [26,] -30.22543477 -29.42302209 -26.0025459 -27.09897707 -26.96222031
## [27,] -15.33553969 -16.17493312 -13.2253797 -14.38786840 -15.14491021
## [28,] -68.19705814 -64.22671273 -58.6320010 -60.35979207 -61.63428334
## [29,]  -8.64766886  -6.13491643  -7.0950003  -6.42459970  -6.21470579
## [30,]  22.51356445  29.47302153  22.4035137  25.07454649  24.45595668
## [31,]  62.22107599  29.66445400  50.8453271  41.16339419  46.72557556
## [32,]  29.66445400  48.86686702  28.9628171  38.56323395  31.45508738
## [33,]  50.84532715  28.96281714  45.8801357  34.80359610  42.56894397
## [34,]  41.16339419  38.56323395  34.8035961  41.31332958  33.69889116
## [35,]  46.72557556  31.45508738  42.5689440  33.69889116  43.37698371
## [36,]  43.16759291  35.70374333  37.7078472  38.29466273  34.52938103
## [37,]  45.53204723  32.98784642  40.6479623  35.74898481  41.39111549
## [38,]  43.63917824  36.29723133  38.9324347  37.61730540  38.55119135
## [39,]  18.18711399  16.57251567  16.9170587  16.37705242  16.96442703
## [40,] 109.58838954  75.03900338  94.8952380  85.54139207  90.33110374
## [41,]   4.42503511  10.71133486   5.1041701   7.76511348   6.26698516
## [42,]  19.69460833  14.90283801  16.3966024  15.59860681  16.46873179
## [43,]  18.31722424  18.73948848  15.5324897  16.75679216  16.80296139
## [44,]  22.81278197  12.11134845  18.3361407  15.54326456  17.72774551
## [45,]  16.93856392  18.66239247  14.3782362  16.30094149  15.94952089
## [46,]  20.17634486  14.94351702  17.4442010  14.40771898  18.20874222
## [47,]  16.95389898  18.42605277  13.8518508  17.11238281  15.34408508
## [48,]  18.21372193  17.24770531  15.1436402  16.34650071  14.25828275
## [49,]  19.75771945  15.14599700  16.6890331  15.04250870  19.98773499
## [50,]  17.45096718  17.95746315  14.9222181  15.99375190  15.33254660
## [51,]   6.57647661   5.60655590   5.2767453   5.46712375   6.44172110
## [52,]  50.99988861  39.84478947  42.7925393  40.67866599  42.53284488
## [53,]   2.57345506  -7.45191463   1.3399468  -3.18938905  -2.96228945
## [54,] -12.24121092 -14.15629874 -11.7398406 -12.63274471 -12.49075763
## [55,] -25.07972564 -16.27396310 -21.4326773 -18.93103993 -20.71284634
## [56,] -15.97700859 -21.93124525 -15.1522488 -18.51371245 -15.82004514
## [57,] -21.50406461 -15.59721463 -19.6600153 -16.60832374 -19.28775375
## [58,] -19.58665886 -18.55792129 -16.7644316 -19.63998240 -16.17584349
## [59,] -19.91654876 -16.60647581 -18.5774689 -16.05844932 -19.74798643
## [60,] -19.45958377 -17.51120313 -17.2087619 -18.04872049 -15.60277505
## [61,] -20.20222542 -17.76793884 -18.5311751 -17.41401776 -20.77779182
## [62,] -19.33027025 -17.98579242 -17.5989084 -17.59989108 -17.48220945
## [63,]  -6.48758796  -6.90832188  -6.1792383  -6.44457853  -6.81614147
## [64,] -54.29770498 -40.07110769 -47.6892729 -43.78893668 -45.78908120
## [65,]  -1.04632674   1.90842344  -1.0647720   0.54493718   0.87669277
## [66,]   0.04059909   0.02396129   0.0350248   0.02948793   0.03205771
##              [,36]        [,37]         [,38]        [,39]         [,40]
##  [1,]   5.36380120   5.54788500   4.369824270   1.87025657   14.41117105
##  [2,]  -1.39323413  -1.24348436  -0.585393444  -0.97071579   -2.09199421
##  [3,]  -7.41545691  -7.90240172  -6.329346949  -2.37628965  -21.00923451
##  [4,]  -4.53064031  -4.02268673  -3.452734125  -1.71042074  -10.30139620
##  [5,]  -5.71191198  -6.48580107  -4.896588179  -1.96829354  -16.68702585
##  [6,]  -5.52328755  -4.58473297  -4.007984739  -1.76014619  -12.98462702
##  [7,]  -3.93497022  -7.30935176  -4.595929968  -1.87673241  -15.17912021
##  [8,]  -7.45971494  -3.42485038  -4.189884167  -1.89201853  -14.21364615
##  [9,]  -4.20933793  -7.99442591  -3.469931996  -2.49839539  -13.61298697
## [10,]  -5.34130476  -4.88701110  -9.068748510   0.74905996  -16.31707249
## [11,]  -1.54626860  -2.20541091   1.007309588  -3.40576982    0.40814271
## [12,] -14.54100521 -14.72318106 -12.184865848  -2.81718407  -47.88154169
## [13,]  -0.06895664   1.24934806  -0.699667118   4.04347305  -13.30869217
## [14,]  27.30478621  27.76702953  30.191968121  13.11874520   62.21155248
## [15,] -37.40114172 -38.08132452 -38.272891290 -16.71592655  -88.83755021
## [16,] -15.90572823 -16.28379172 -17.578664664  -7.19174036  -37.63523575
## [17,]  17.62405756  17.84223764  18.290731429   8.32094039   40.46558578
## [18,] -25.34003212 -25.89558238 -28.004099362 -12.03786027  -58.37672059
## [19,] -28.41329891 -28.31897840 -31.459267925 -13.64775736  -63.26255210
## [20,] -26.60349363 -28.22964345 -29.313164168 -12.84879106  -63.45843639
## [21,] -27.72327689 -27.48905549 -30.690829132 -13.18688225  -61.74005619
## [22,] -26.85884213 -28.19716819 -30.604034432 -12.90276413  -63.12235511
## [23,] -27.59951717 -26.99190353 -29.496401370 -13.45792765  -60.41379247
## [24,] -30.91847622 -23.18520368 -31.274690399 -12.89481309  -61.86734450
## [25,] -22.88965122 -33.09666888 -27.787017441 -13.74360489  -62.47481664
## [26,] -29.36955246 -25.85391045 -37.143521217 -10.03123080  -62.27703438
## [27,] -11.39079823 -15.92377088 -16.899585057  -9.68112412  -31.72794841
## [28,] -62.05011643 -59.98776414 -57.547106777 -31.54821910 -133.54244187
## [29,] -12.15445987  -4.11376722   7.639555310  -6.43055613   -8.93708545
## [30,]  24.64542038  24.30220669  26.039881097  14.07287201   48.69509097
## [31,]  43.16759290  45.53204723  43.639178237  18.18711399  109.58838955
## [32,]  35.70374333  32.98784642  36.297231327  16.57251567   75.03900338
## [33,]  37.70784715  40.64796232  38.932434723  16.91705875   94.89523800
## [34,]  38.29466273  35.74898481  37.617305402  16.37705242   85.54139207
## [35,]  34.52938103  41.39111549  38.551191346  16.96442703   90.33110374
## [36,]  44.74133891  30.24907466  34.969371452  17.95064231   86.88805987
## [37,]  30.24907466  53.50579137  37.106992212  17.95998186   89.03300877
## [38,]  34.96937145  37.10699221  72.766719538   0.11985375   89.63525390
## [39,]  17.95064231  17.95998186   0.119853751  22.36091226   28.44287349
## [40,]  86.88805987  89.03300877  89.635253901  28.44287349  258.26810285
## [41,]   5.68339819   5.35516666  21.308417726 -14.70624261   81.45070999
## [42,]  15.69585456  16.18486000  16.952886150   6.27861082   39.53811111
## [43,]  16.44481625  16.61850647  18.266928336   7.67500908   37.74575087
## [44,]  16.10250276  17.16180166  17.436272615   7.01844946   40.68355037
## [45,]  15.82783341  15.86800682  17.613146512   7.26229450   36.22350325
## [46,]  15.92736152  16.54501828  18.753158751   6.86815205   38.97290008
## [47,]  15.43639195  16.27170355  15.969387736   7.68767850   35.76078267
## [48,]  22.09384828   8.27637480  20.378849209   6.63817077   36.48465159
## [49,]   6.47259201  31.17644252   3.912411488  11.50795186   38.50752483
## [50,]  16.74504392  14.85593113  51.180194070  -5.99738821   33.99802319
## [51,]   2.78951926   7.02329381  -8.513482253  10.83951305   20.21631093
## [52,]  49.44702015  39.83462824   3.455571099  42.71533085   45.51935600
## [53,]  13.91019522  -6.84945698 -62.819571919  33.82107144  -96.99848941
## [54,] -12.47741757 -12.39812978 -13.163582939  -6.86083795  -25.59482095
## [55,] -19.50643517 -20.23220353 -20.173654559  -8.90131640  -46.81595711
## [56,] -17.38335470 -16.43910723 -17.672131728  -8.30911073  -36.98954359
## [57,] -17.55893589 -18.48886420 -18.542361498  -8.35484643  -41.78450341
## [58,] -18.07759833 -17.35313936 -17.501128908  -8.35768529  -40.08837606
## [59,] -16.38584903 -18.79173262 -19.359672443  -8.06461748  -40.29159417
## [60,] -23.19907478 -11.84251590 -12.044813235 -10.94642423  -38.43100510
## [61,]  -9.35243559 -29.78126121 -32.613475469  -2.66022501  -43.73505015
## [62,] -17.68918632 -16.89772694 -38.097966812   0.17723508  -37.81076238
## [63,]  -5.24766850  -7.36950697   1.398106574  -9.14681797  -15.97420039
## [64,] -48.89222936 -44.17273963 -23.409117400 -31.06563512  -90.26163628
## [65,]  -8.06812038   2.99766431  33.165291489 -16.86337213   33.67233577
## [66,]   0.03553578   0.03057162   0.006151924   0.02548176    0.05683107
##               [,41]        [,42]        [,43]        [,44]         [,45]
##  [1,]   -2.86658888  -7.61580769 -10.81463740  -8.75305173 -10.432200160
##  [2,]    1.49501734   8.62992481  10.35190179   9.51032175  10.004887148
##  [3,]    3.62797769   6.98914328  10.87800005   7.98796127  10.573272915
##  [4,]    1.98885872   7.82260262  10.22989458   9.48332304   9.739540569
##  [5,]    3.40716616   7.53563556  11.15431317   8.49726483  10.862393981
##  [6,]    2.47826809   7.73039147  10.70398248   8.99399275  10.273448476
##  [7,]    3.43779703   7.50897004  10.86604296   8.51454171  10.529730931
##  [8,]    2.23662092   7.69703837  10.83750685   8.88489027  10.444506913
##  [9,]    5.14936109   7.61109069  10.85806047   8.72201210  10.479053350
## [10,]   -2.87023716   7.55016428  10.65656656   8.70336526  10.273137925
## [11,]   10.06620300   5.01507560   6.54157803   5.65400168   6.306757367
## [12,]   -6.34666621  13.91988076  21.23798214  16.25108420  20.499197511
## [13,]  -28.99290591   3.77483084   4.42018318   4.11867165   4.275797518
## [14,]   12.54367224  31.87525723  41.09383245  36.76761580  39.088870576
## [15,]   -7.00023096 -16.09256761 -16.67873866 -16.83483046 -15.970510104
## [16,]   -6.62406177 -19.22635114 -24.79029628 -22.45226409 -23.518693759
## [17,]    4.65921773   9.30263171  10.09741696   9.59491901   9.747694626
## [18,]  -11.32771499 -30.16595715 -37.38269946 -33.98407998 -35.780728453
## [19,]  -13.45880768 -32.39333937 -42.67804153 -36.67364926 -40.583901513
## [20,]  -11.30368321 -31.74503306 -39.69298918 -38.43582481 -37.457613860
## [21,]  -12.98610753 -31.86951647 -41.54632962 -36.02529432 -39.789080329
## [22,]  -12.64981251 -32.01429820 -40.89642288 -37.54946305 -38.600979587
## [23,]  -12.30779497 -31.77494475 -41.46049716 -36.11209149 -39.720472622
## [24,]  -12.39986675 -31.81048709 -40.96951784 -36.80317839 -38.897824276
## [25,]  -11.85723602 -31.96876669 -41.19689098 -36.83749310 -39.233727035
## [26,]  -16.32166860 -31.65893308 -40.85472958 -36.57611685 -38.818028605
## [27,]   -5.13728284 -17.60383116 -22.71043218 -20.05045991 -21.720468719
## [28,]  -17.39311684 -66.94985073 -86.36651385 -77.63783462 -81.972621082
## [29,]   10.39292738  -9.09126373 -11.14588328 -10.76320513 -10.499504100
## [30,]   11.22072878  10.42673577  16.17707723  12.31422355  15.312352693
## [31,]    4.42503511  19.69460833  18.31722424  22.81278197  16.938563917
## [32,]   10.71133486  14.90283801  18.73948848  12.11134845  18.662392466
## [33,]    5.10417007  16.39660236  15.53248966  18.33614069  14.378236185
## [34,]    7.76511348  15.59860681  16.75679216  15.54326456  16.300941492
## [35,]    6.26698516  16.46873179  16.80296139  17.72774551  15.949520891
## [36,]    5.68339819  15.69585456  16.44481625  16.10250276  15.827833411
## [37,]    5.35516666  16.18486000  16.61850647  17.16180166  15.868006820
## [38,]   21.30841773  16.95288615  18.26692834  17.43627261  17.613146512
## [39,]  -14.70624261   6.27861082   7.67500908   7.01844946   7.262294499
## [40,]   81.45070999  39.53811111  37.74575087  40.68355037  36.223503253
## [41,]  143.88144816   4.34075052   7.58968612   5.19789796   7.237445474
## [42,]    4.34075052  17.52160533  20.03303791  19.17921522  19.239343511
## [43,]    7.58968612  20.03303791  26.78209465  23.44866496  25.351930362
## [44,]    5.19789796  19.17921522  23.44866496  23.67717620  21.943878625
## [45,]    7.23744547  19.23934351  25.35193036  21.94387862  24.264901403
## [46,]    6.79503047  19.39287080  24.65427284  23.06471180  23.157275900
## [47,]    6.45534779  19.09790062  25.08155813  21.90192951  24.031219734
## [48,]    6.04650217  19.15593089  24.80036823  22.30440991  23.513676843
## [49,]    3.62746767  18.96897001  24.14486978  22.43506564  22.862711311
## [50,]   12.01181690  19.63948677  25.73994229  22.65093357  24.489599497
## [51,]    8.74897770   7.91968632  10.19658416   9.32102581   9.697233360
## [52,]  -92.95808538  44.01007085  55.56907621  51.98576820  52.411297880
## [53,] -191.34860363  -3.66653496  -6.41801231  -3.00153515  -6.589429539
## [54,]   -5.29759247  -6.38806370  -8.54499301  -7.05078736  -8.168622248
## [55,]   -4.18627607 -10.64961597 -10.99394295 -11.62791222 -10.455512911
## [56,]   -5.47757751  -8.89181529 -10.53149530  -8.07959879 -10.320227041
## [57,]   -4.33767046  -9.43204901  -9.91803407 -10.07473107  -9.465458731
## [58,]   -4.25542525  -9.02918376  -9.73838728  -9.12566911  -9.439589606
## [59,]   -5.37830997  -9.53343815 -10.53977114  -9.89195500 -10.192352765
## [60,]   -0.61761152  -8.87338955  -9.57833618  -9.04581838  -9.253911751
## [61,]  -13.53495910 -10.06236570 -11.23062224 -10.37371523 -10.893896494
## [62,]   -7.34713796  -9.53571556 -10.63760037  -9.68898232 -10.308716213
## [63,]   -4.22417863  -2.77410416  -3.44003596  -2.93257698  -3.320370219
## [64,]   29.96119654 -23.94428489 -24.11782718 -24.79954358 -23.178414599
## [65,]   66.39803276   3.24068641   4.36568526   3.03204762   4.411430675
## [66,]   -0.03888437   0.01172314   0.01063061   0.01297802   0.009886694
##              [,46]        [,47]        [,48]        [,49]         [,50]
##  [1,]  -9.70340843 -10.34454425 -10.01935669  -9.38808607 -10.733366052
##  [2,]   9.85384600   9.94594849   9.85860838   9.59458354  10.301237754
##  [3,]   9.38249886  10.47058831   9.96507361   9.03957676  10.856786632
##  [4,]   9.84115083   9.63137229   9.58313625   9.47903435  10.169844010
##  [5,]   9.59354725  10.87864666  10.37308281   9.28979779  11.119702623
##  [6,]   9.97547949  10.04727701   9.58628884  10.04360864  10.382240371
##  [7,]   9.46267227  10.45191908  11.54269584   6.91696206  11.883464118
##  [8,]   9.74568287  10.50710559   8.05269323  12.54058966   9.445969320
##  [9,]   9.77105805  10.25759288  11.30407858   6.96676764  11.899485740
## [10,]   9.43952769  10.41003259   9.20081821  11.45995126   5.276705317
## [11,]   6.09816789   6.18509977   6.35281032   5.23551402   8.439999235
## [12,]  18.59670932  20.30988540  19.50452267  17.76131687  21.575719263
## [13,]   4.19868710   4.30599631   4.12264878   4.27440111   3.838927752
## [14,]  38.36495846  38.69054035  38.38894701  37.44696957  39.833092903
## [15,] -16.60500867 -15.77919754 -16.08901572 -16.25256101 -16.161213052
## [16,] -23.24220447 -23.29841720 -23.17307342 -22.71517458 -23.967178992
## [17,]   9.79623176   9.62394471   9.66966434   9.54053463   9.901604157
## [18,] -35.29346920 -35.44405526 -35.20351638 -34.43156148 -36.507799298
## [19,] -39.02507154 -40.07107984 -39.53275706 -38.12755587 -41.209608704
## [20,] -38.48246504 -37.27974635 -37.48001203 -37.47892220 -38.510740982
## [21,] -38.02150693 -39.27716597 -38.70844746 -37.16614914 -40.380234529
## [22,] -40.15574716 -37.39736016 -38.61173517 -37.29451418 -40.501478799
## [23,] -37.12483046 -40.28778121 -38.46148227 -37.40850313 -39.391202535
## [24,] -38.79983506 -37.79676097 -43.71312814 -28.00714886 -45.215662034
## [25,] -37.95192892 -39.63118285 -31.48618257 -50.66469791 -30.438852936
## [26,] -38.68311871 -37.66296030 -43.14877445 -27.04863607 -58.536636423
## [27,] -21.07173568 -21.58466412 -18.65875824 -23.55052525 -16.146238677
## [28,] -80.40676313 -81.55065371 -80.75712676 -81.84942632 -77.412237761
## [29,] -10.27975858 -11.06409398 -12.53087652 -13.62023038  -4.568202561
## [30,]  14.07179613  14.98309594  14.50892997  13.65008916  15.403424786
## [31,]  20.17634486  16.95389898  18.21372193  19.75771945  17.450967177
## [32,]  14.94351702  18.42605277  17.24770531  15.14599700  17.957463154
## [33,]  17.44420097  13.85185085  15.14364022  16.68903311  14.922218067
## [34,]  14.40771898  17.11238281  16.34650071  15.04250870  15.993751904
## [35,]  18.20874222  15.34408508  14.25828275  19.98773499  15.332546597
## [36,]  15.92736152  15.43639195  22.09384828   6.47259201  16.745043924
## [37,]  16.54501827  16.27170355   8.27637480  31.17644252  14.855931131
## [38,]  18.75315875  15.96938774  20.37884921   3.91241149  51.180194069
## [39,]   6.86815205   7.68767850   6.63817077  11.50795186  -5.997388206
## [40,]  38.97290008  35.76078267  36.48465159  38.50752483  33.998023193
## [41,]   6.79503047   6.45534779   6.04650217   3.62746767  12.011816902
## [42,]  19.39287080  19.09790062  19.15593089  18.96897001  19.639486775
## [43,]  24.65427284  25.08155813  24.80036823  24.14486978  25.739942286
## [44,]  23.06471180  21.90192951  22.30440991  22.43506564  22.650933567
## [45,]  23.15727590  24.03121973  23.51367684  22.86271131  24.489599497
## [46,]  25.06426348  21.91432923  23.74149047  21.74602732  25.351803813
## [47,]  21.91432923  24.84949295  22.34534674  24.56288922  22.147424914
## [48,]  23.74149047  22.34534674  32.08636428   6.37108249  33.629957233
## [49,]  21.74602732  24.56288922   6.37108249  55.94526873  -2.167542825
## [50,]  25.35180381  22.14742491  33.62995723  -2.16754282  78.517441876
## [51,]   8.94841258  10.54221500   3.09956167  23.11673675 -20.789948488
## [52,]  51.32794621  53.92797236  51.99504554  66.13532904  22.872384540
## [53,]  -6.86840112  -3.72563351  -1.34883507  10.40660108 -37.110087201
## [54,]  -7.72643138  -8.02623597  -7.86484082  -7.50425020  -8.256681388
## [55,] -11.23609623 -10.36361413 -10.67674393 -10.89510223 -10.751289545
## [56,]  -8.99209675 -10.26300259  -9.83450272  -9.10572294 -10.047505293
## [57,] -10.28543074  -9.16709858  -9.55995289  -9.70220349  -9.870435773
## [58,]  -8.18697697 -10.08278698  -9.15915520  -9.47973723  -8.636472581
## [59,] -10.98494845  -9.67105957  -9.50096390 -10.57785603 -11.096699715
## [60,]  -9.11168913  -9.05492544 -14.31988527  -2.06414745  -7.397575175
## [61,] -10.93095486 -10.77075418  -3.08649995 -20.39816199 -20.189383766
## [62,] -10.90979317  -9.08420805 -14.70141554   2.93127080 -38.698834468
## [63,]  -2.87725942  -3.68415022  -0.36605848  -9.22590035  11.245961785
## [64,] -23.50665792 -23.99252085 -23.11682158 -31.87454987  -5.613448180
## [65,]   4.83865638   2.88147323   2.66625403  -5.95821008  24.848713834
## [66,]   0.01081028   0.01095106   0.01026279   0.02036393  -0.009209929
##               [,51]       [,52]        [,53]         [,54]        [,55]
##  [1,]  -4.468201765  -19.665508    4.3604121   1.534652822   0.36087397
##  [2,]   4.385318660   20.761225   -2.9720632  -1.452967359  -2.30520057
##  [3,]   4.416550536   18.589460   -5.3027477  -1.596932821   0.83290584
##  [4,]   4.380063461   19.760808   -3.2870604  -1.225519804  -1.19873396
##  [5,]   4.516162380   19.727293   -4.9031496  -1.701071509   0.04951674
##  [6,]   4.640521011   19.785555   -4.2737454  -1.483784665  -0.68255960
##  [7,]   3.583008387   20.019620   -3.0220665  -1.573599304  -0.11630602
##  [8,]   5.567259481   18.976201   -6.5166717  -1.537114888  -0.49290475
##  [9,]   4.000819996   18.445127   -6.1890289  -1.551956239  -0.30669926
## [10,]   6.731023521   25.637822    4.6772566  -1.488946193  -0.40104601
## [11,]   1.402010383    7.406292  -10.0076326  -0.896359329  -0.86298067
## [12,]   8.181139984   44.272949    3.1090826  -3.088446802   0.93386334
## [13,]   1.915534073   25.164608   26.2578891  -0.687236204  -1.28545966
## [14,]  15.629162956   86.567503   -9.9525125 -14.342574806 -18.96774971
## [15,]  -5.472695204  -42.067407    1.8705387  12.496797386  20.02330368
## [16,]  -9.560355227  -52.318998    5.4505508   7.830511621  10.80696903
## [17,]   3.370511739   24.040202   -2.1985309  -7.055059594 -10.40701698
## [18,] -14.434441050  -79.326906    8.8645475  13.237923895  18.13116721
## [19,] -15.935421420  -88.385912   11.5485465  15.141059894  18.92320997
## [20,] -15.489666539  -86.378278    7.2687274  13.526704522  19.67238041
## [21,] -15.619946591  -86.194976   11.3965973  14.653930545  18.65829305
## [22,] -15.305752032  -86.395833   10.0469503  14.175889004  19.38298623
## [23,] -15.941750957  -87.211114    9.5889310  14.598490163  18.52605198
## [24,] -11.707810005  -87.441080    5.3456297  14.277699591  18.94996301
## [25,] -22.023935581  -86.485855   14.9755756  14.387229949  19.05504851
## [26,]  -2.614446306  -84.626878    3.1996890  14.227008757  18.77600892
## [27,] -21.833549961  -32.621304   40.0821397   7.842250500  10.06777233
## [28,] -30.278234950 -211.507804  -29.8353263  30.450895171  40.83407243
## [29,]   8.804597237  -82.881751 -107.7878181   3.574841642   5.22571818
## [30,]   4.939051255   33.961549   -5.6951309 -12.189586982 -12.57568273
## [31,]   6.576476608   50.999889    2.5734551 -12.241210918 -25.07972564
## [32,]   5.606555898   39.844789   -7.4519146 -14.156298736 -16.27396310
## [33,]   5.276745288   42.792539    1.3399468 -11.739840607 -21.43267730
## [34,]   5.467123755   40.678666   -3.1893890 -12.632744705 -18.93103993
## [35,]   6.441721098   42.532845   -2.9622895 -12.490757628 -20.71284634
## [36,]   2.789519263   49.447020   13.9101952 -12.477417569 -19.50643517
## [37,]   7.023293813   39.834628   -6.8494570 -12.398129782 -20.23220353
## [38,]  -8.513482253    3.455571  -62.8195719 -13.163582939 -20.17365456
## [39,]  10.839513047   42.715331   33.8210714  -6.860837946  -8.90131640
## [40,]  20.216310932   45.519356  -96.9984894 -25.594820946 -46.81595711
## [41,]   8.748977696  -92.958085 -191.3486036  -5.297592466  -4.18627607
## [42,]   7.919686320   44.010071   -3.6665350  -6.388063703 -10.64961597
## [43,]  10.196584161   55.569076   -6.4180123  -8.544993006 -10.99394295
## [44,]   9.321025810   51.985768   -3.0015351  -7.050787364 -11.62791222
## [45,]   9.697233360   52.411298   -6.5894295  -8.168622248 -10.45551291
## [46,]   8.948412583   51.327946   -6.8684011  -7.726431378 -11.23609623
## [47,]  10.542214998   53.927972   -3.7256335  -8.026235968 -10.36361413
## [48,]   3.099561669   51.995046   -1.3488351  -7.864840821 -10.67674393
## [49,]  23.116736754   66.135329   10.4066011  -7.504250199 -10.89510223
## [50,] -20.789948488   22.872385  -37.1100872  -8.256681388 -10.75128955
## [51,]  34.060739319   14.800029  -34.2698802  -2.711759882  -3.82169791
## [52,]  14.800029459  279.455798  265.5649656 -18.651052361 -27.35996372
## [53,] -34.269880209  265.564966  497.7461031   2.812586044   1.25575294
## [54,]  -2.711759882  -18.651052    2.8125860   6.188430786   7.22379802
## [55,]  -3.821697912  -27.359964    1.2557529   7.223798024  12.05959148
## [56,]  -3.390142508  -23.370063    3.2606136   7.439428186   9.29455576
## [57,]  -3.313570628  -24.241047    1.7529066   6.922621916  10.79741877
## [58,]  -3.555304996  -24.048590    0.8242367   6.881335110  10.08226031
## [59,]  -3.538521758  -22.837513    5.9131735   7.277117213  10.56487575
## [60,]  -1.292596230  -36.445853  -22.3496454   6.884022140  10.09022695
## [61,]  -4.725347276    3.770043   54.3896571   7.462549337  10.75282328
## [62,]  12.804007615   -8.308461   19.2332810   7.305930770  10.37709022
## [63,] -15.754080672   -4.652863   16.9728980   3.075818945   3.72848411
## [64,]  -7.765265412 -130.429987 -117.2976577  15.898230361  26.40459764
## [65,]  13.059489877 -114.563093 -218.5537178  -0.335619971  -0.23865515
## [66,]   0.001243344    0.108348    0.1359816  -0.008509881  -0.01619920
##              [,56]        [,57]        [,58]       [,59]        [,60]
##  [1,]   1.41704989   0.47055267   0.49601537   1.1604498   0.54544734
##  [2,]  -2.04685621  -1.96505706  -1.81276128  -2.2388056  -1.77516077
##  [3,]  -1.28501216   0.37743091   0.15829432  -0.6067165   0.06599072
##  [4,]  -0.69068667  -1.12040407  -0.50221539  -1.4871568  -0.65318758
##  [5,]  -1.68297878  -0.03978155  -0.57672919  -0.9413140  -0.55744813
##  [6,]  -1.14944152  -0.88094279  -0.21075556  -1.5514827  -0.30723498
##  [7,]  -1.64320022  -0.16938782  -0.71842538  -0.6337836  -1.65038830
##  [8,]  -1.34359606  -0.62541385  -0.45455378  -1.5433080   0.96986867
##  [9,]  -1.47366230  -0.40999859  -0.50802938  -1.0027874  -1.30314183
## [10,]  -1.38444851  -0.48567974  -0.57575386  -1.0814164  -1.01260759
## [11,]  -1.07352796  -0.77377342  -0.71511263  -1.0616175  -0.65071213
## [12,]  -2.23907166   0.32302290   0.12865899  -1.3925234   0.05707080
## [13,]  -1.06769834  -1.09737241  -1.01149377  -1.2402769  -0.88542335
## [14,] -17.67153189 -17.25735414 -16.71217776 -18.1619376 -16.54834486
## [15,]  16.90781355  18.20932520  17.62995144  17.8766243  17.45573754
## [16,]   9.68979579   9.68407299   9.30385038  10.1277069   9.19086733
## [17,]  -9.34291949  -9.61126523  -9.37014650  -9.6629632  -9.30962349
## [18,]  16.58203747  16.45239638  15.80886995  17.2668624  15.70608710
## [19,]  19.21750231  17.52037325  17.51511428  18.6606229  17.19709034
## [20,]  15.40656316  17.59970623  15.95035160  17.9930072  16.04424628
## [21,]  18.61100964  17.04448440  17.12445500  18.1273297  16.78650914
## [22,]  16.72761365  17.80598061  15.54270313  18.9433921  16.26907692
## [23,]  18.41562686  16.77641907  17.49941233  17.7734453  16.89212420
## [24,]  17.52862062  17.24593322  16.45073352  17.7845087  19.65827083
## [25,]  17.72575201  17.33297483  16.95252314  18.6374605  12.69949359
## [26,]  17.43639154  17.12332127  16.26665129  17.8945099  18.15473300
## [27,]   9.63878059   9.20840721   8.93246930  10.2089790   5.72228010
## [28,]  37.66617487  37.05119434  36.10231331  38.2924127  38.68322521
## [29,]   4.34562349   4.59322078   4.64952499   3.6347552  11.91590690
## [30,] -13.87280520 -12.29979549 -12.29689970 -13.2435800 -12.38585217
## [31,] -15.97700859 -21.50406461 -19.58665886 -19.9165488 -19.45958377
## [32,] -21.93124525 -15.59721463 -18.55792129 -16.6064758 -17.51120313
## [33,] -15.15224879 -19.66001532 -16.76443159 -18.5774689 -17.20876192
## [34,] -18.51371245 -16.60832374 -19.63998240 -16.0584493 -18.04872049
## [35,] -15.82004514 -19.28775375 -16.17584349 -19.7479864 -15.60277505
## [36,] -17.38335470 -17.55893589 -18.07759833 -16.3858490 -23.19907478
## [37,] -16.43910723 -18.48886420 -17.35313936 -18.7917326 -11.84251590
## [38,] -17.67213173 -18.54236150 -17.50112891 -19.3596724 -12.04481323
## [39,]  -8.30911073  -8.35484643  -8.35768529  -8.0646175 -10.94642423
## [40,] -36.98954359 -41.78450341 -40.08837606 -40.2915942 -38.43100510
## [41,]  -5.47757751  -4.33767046  -4.25542525  -5.3783100  -0.61761152
## [42,]  -8.89181529  -9.43204901  -9.02918376  -9.5334381  -8.87338955
## [43,] -10.53149530  -9.91803407  -9.73838728 -10.5397711  -9.57833618
## [44,]  -8.07959879 -10.07473107  -9.12566911  -9.8919550  -9.04581838
## [45,] -10.32022704  -9.46545873  -9.43958961 -10.1923528  -9.25391175
## [46,]  -8.99209675 -10.28543074  -8.18697697 -10.9849484  -9.11168913
## [47,] -10.26300259  -9.16709858 -10.08278698  -9.6710596  -9.05492544
## [48,]  -9.83450272  -9.55995289  -9.15915520  -9.5009639 -14.31988527
## [49,]  -9.10572294  -9.70220349  -9.47973723 -10.5778560  -2.06414745
## [50,] -10.04750529  -9.87043577  -8.63647258 -11.0966997  -7.39757518
## [51,]  -3.39014251  -3.31357063  -3.55530500  -3.5385218  -1.29259623
## [52,] -23.37006311 -24.24104711 -24.04858963 -22.8375129 -36.44585299
## [53,]   3.26061361   1.75290661   0.82423669   5.9131735 -22.34964536
## [54,]   7.43942819   6.92262192   6.88133511   7.2771172   6.88402214
## [55,]   9.29455576  10.79741877  10.08226031  10.5648758  10.09022695
## [56,]  11.03731988   8.75196083   9.93124456   9.0151597   9.40329428
## [57,]   8.75196083  10.07879317   8.92573170  10.0551814   9.19798300
## [58,]   9.93124456   8.92573170  10.57128167   8.5249173   9.53346520
## [59,]   9.01515974  10.05518142   8.52491730  10.7372683   8.47703363
## [60,]   9.40329428   9.19798300   9.53346520   8.4770336  14.56651504
## [61,]   9.60543584  10.07773319   9.36550357  11.3999981   0.42494503
## [62,]   9.57805395   9.71073619   9.04997141  10.2422196   7.85953781
## [63,]   3.73673235   3.54537883   3.67863402   3.6899809   2.76913514
## [64,]  22.08306082  23.87178007  23.51576035  22.4413326  29.77216355
## [65,]  -0.70264131  -0.31231480   0.30383283  -2.3440527  12.02153827
## [66,]  -0.01178622  -0.01428646  -0.01388322  -0.0122949  -0.02104195
##               [,61]        [,62]         [,63]         [,64]         [,65]
##  [1,]   1.398485892   1.21046414   0.319353424    0.29783432   -2.80950112
##  [2,]  -2.519315506  -2.25401047  -0.419379264   -4.85250938    2.52349717
##  [3,]  -0.820063568  -0.70983594  -0.271416494    2.15412232    3.03638984
##  [4,]  -1.664448300  -1.29832800  -0.256187050   -1.72651297    2.49206404
##  [5,]  -1.178312474  -1.17248367  -0.340418547    0.47391176    2.95353856
##  [6,]  -1.898448023  -1.18406829  -0.402094188   -0.81571736    2.86111320
##  [7,]   0.203687910  -1.63568242   0.066440677   -0.24979299    1.99814206
##  [8,]  -3.523129227  -0.69718839  -0.812262602    0.01096621    4.05794207
##  [9,]   0.006643482  -1.82384858  -0.030083392    0.18537587    3.30066582
## [10,]  -0.037201541   1.78902337  -1.613234053   -3.18498218   -1.11518576
## [11,]  -1.502510999  -2.26367674   0.781671838   -0.10749009    3.95750444
## [12,]  -1.986500276  -1.81944091  -0.614716155    1.58345253    2.10950280
## [13,]  -1.429042626  -0.91089101  -0.714374360   -7.53886685   -6.26210179
## [14,] -19.166424933 -18.21573252  -6.147114754  -41.45782536    6.24466357
## [15,]  18.299084905  17.81321303   6.401459854   45.30092801   -0.11857121
## [16,]  10.730755715  10.13813401   3.192087799   23.69848309   -3.96736832
## [17,]  -9.911897350  -9.65634933  -3.603100900  -23.38242511    0.43674108
## [18,]  18.143782018  17.23654066   5.780562560   39.64783805   -5.68389550
## [19,]  19.865278417  19.00826577   6.450541387   42.20673102   -6.89088737
## [20,]  18.693391490  17.49403634   5.870384867   41.93658395   -5.13773343
## [21,]  19.332853854  18.56112659   6.243424669   41.10686740   -6.90194958
## [22,]  19.337041997  18.51877724   5.924633316   41.53758042   -6.36586266
## [23,]  18.760341552  17.91389070   6.374327232   41.26232123   -6.01770489
## [24,]  14.311337245  20.68013751   4.399047131   41.79427080   -4.22448202
## [25,]  25.008866540  13.86928890   8.980091048   41.56259891   -8.40079806
## [26,]  17.674005663  27.31402948   0.054393109   40.19993084   -3.30580031
## [27,]  16.049921653   6.88631315  10.029456694   14.56476027  -20.96013994
## [28,]  33.975348363  35.66709581  11.613736679  103.36416120   11.98167297
## [29,] -10.846214804   1.24035376  -5.664823668   40.08575353   53.00660566
## [30,] -13.529636211 -13.28842582  -5.841817358  -27.58399387    0.51930100
## [31,] -20.202225424 -19.33027025  -6.487587958  -54.29770498   -1.04632674
## [32,] -17.767938844 -17.98579242  -6.908321882  -40.07110769    1.90842344
## [33,] -18.531175092 -17.59890843  -6.179238275  -47.68927291   -1.06477201
## [34,] -17.414017759 -17.59989108  -6.444578529  -43.78893668    0.54493718
## [35,] -20.777791819 -17.48220945  -6.816141465  -45.78908120    0.87669277
## [36,]  -9.352435594 -17.68918632  -5.247668504  -48.89222936   -8.06812038
## [37,] -29.781261206 -16.89772694  -7.369506969  -44.17273963    2.99766431
## [38,] -32.613475469 -38.09796681   1.398106574  -23.40911740   33.16529149
## [39,]  -2.660225007   0.17723508  -9.146817970  -31.06563512  -16.86337213
## [40,] -43.735050148 -37.81076238 -15.974200389  -90.26163628   33.67233577
## [41,] -13.534959103  -7.34713796  -4.224178634   29.96119654   66.39803276
## [42,] -10.062365696  -9.53571556  -2.774104162  -23.94428489    3.24068641
## [43,] -11.230622242 -10.63760037  -3.440035957  -24.11782718    4.36568526
## [44,] -10.373715232  -9.68898232  -2.932576981  -24.79954358    3.03204762
## [45,] -10.893896494 -10.30871621  -3.320370219  -23.17841460    4.41143068
## [46,] -10.930954856 -10.90979317  -2.877259424  -23.50665792    4.83865638
## [47,] -10.770754178  -9.08420805  -3.684150216  -23.99252085    2.88147323
## [48,]  -3.086499954 -14.70141554  -0.366058479  -23.11682158    2.66625403
## [49,] -20.398161994   2.93127080  -9.225900346  -31.87454987   -5.95821008
## [50,] -20.189383766 -38.69883447  11.245961785   -5.61344818   24.84871383
## [51,]  -4.725347276  12.80400762 -15.754080672   -7.76526541   13.05948988
## [52,]   3.770042878  -8.30846081  -4.652862731 -130.42998663 -114.56309259
## [53,]  54.389657108  19.23328100  16.972897974 -117.29765768 -218.55371779
## [54,]   7.462549337   7.30593077   3.075818945   15.89823036   -0.33561997
## [55,]  10.752823281  10.37709022   3.728484106   26.40459764   -0.23865515
## [56,]   9.605435842   9.57805395   3.736732353   22.08306082   -0.70264131
## [57,]  10.077733186   9.71073619   3.545378830   23.87178007   -0.31231480
## [58,]   9.365503566   9.04997141   3.678634019   23.51576035    0.30383283
## [59,]  11.399998127  10.24221955   3.689980878   22.44133263   -2.34405271
## [60,]   0.424945034   7.85953781   2.769135144   29.77216355   12.02153827
## [61,]  28.862916529  15.68335028   4.132874537    8.72821310  -27.16498763
## [62,]  15.683350277  25.22233704  -4.397237345   13.45802920  -11.87854158
## [63,]   4.132874537  -4.39723735   9.162749329    7.72351886   -5.71824401
## [64,]   8.728213095  13.45802920   7.723518858   92.82139968   55.10907976
## [65,] -27.164987626 -11.87854157  -5.718244014   55.10907976  101.76728497
## [66,]   0.003386987  -0.00252104  -0.003289057   -0.07267699   -0.06354741
##               [,66]
##  [1,]  0.0081103366
##  [2,] -0.0033054483
##  [3,] -0.0108417785
##  [4,] -0.0061453221
##  [5,] -0.0091968864
##  [6,] -0.0074560466
##  [7,] -0.0082288412
##  [8,] -0.0084713360
##  [9,] -0.0087623221
## [10,] -0.0047372114
## [11,] -0.0049553960
## [12,] -0.0183292626
## [13,]  0.0046567713
## [14,]  0.0177716929
## [15,] -0.0314170618
## [16,] -0.0108620200
## [17,]  0.0135729154
## [18,] -0.0164816472
## [19,] -0.0174713145
## [20,] -0.0194526259
## [21,] -0.0169603788
## [22,] -0.0181289659
## [23,] -0.0172400804
## [24,] -0.0182767013
## [25,] -0.0177667001
## [26,] -0.0167370821
## [27,]  0.0005567313
## [28,] -0.0557844685
## [29,] -0.0374930898
## [30,]  0.0168958609
## [31,]  0.0405990919
## [32,]  0.0239612917
## [33,]  0.0350247971
## [34,]  0.0294879294
## [35,]  0.0320577097
## [36,]  0.0355357775
## [37,]  0.0305716202
## [38,]  0.0061519236
## [39,]  0.0254817563
## [40,]  0.0568310686
## [41,] -0.0388843688
## [42,]  0.0117231375
## [43,]  0.0106306073
## [44,]  0.0129780228
## [45,]  0.0098866941
## [46,]  0.0108102803
## [47,]  0.0109510604
## [48,]  0.0102627875
## [49,]  0.0203639287
## [50,] -0.0092099291
## [51,]  0.0012433444
## [52,]  0.1083480123
## [53,]  0.1359816294
## [54,] -0.0085098810
## [55,] -0.0161991982
## [56,] -0.0117862219
## [57,] -0.0142864643
## [58,] -0.0138832192
## [59,] -0.0122949006
## [60,] -0.0210419472
## [61,]  0.0033869869
## [62,] -0.0025210397
## [63,] -0.0032890570
## [64,] -0.0726769904
## [65,] -0.0635474062
## [66,]  0.0050834145
## 
## $log_evidence
## [1] -367.8081
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 233

3d)

A function is defined for you in the code chunk below. This function creates a coefficient summary plot in the style of the coefplot() function, but uses the Bayesian results from the Laplace Approximation. The first argument is the vector of posterior means, and the second argument is the vector of posterior standard deviations. The third argument is the name of the feature associated with each coefficient.

viz_post_coefs <- function(post_means, post_sds, xnames)
{
  tibble::tibble(
    mu = post_means,
    sd = post_sds,
    x = xnames
  ) %>% 
    mutate(x = factor(x, levels = xnames)) %>% 
    ggplot(mapping = aes(x = x)) +
    geom_hline(yintercept = 0, color = 'grey', linetype = 'dashed') +
    geom_point(mapping = aes(y = mu)) +
    geom_linerange(mapping = aes(ymin = mu - 2 * sd,
                                 ymax = mu + 2 * sd,
                                 group = x)) +
    labs(x = 'feature', y = 'coefficient value') +
    coord_flip() +
    theme_bw()
}

Create the posterior summary visualization figure for model 3 and model 6. You must provide the posterior means and standard deviations of the regression coefficients (the \(\beta\) parameters). Do NOT include the \(\varphi\) parameter. The feature names associated with the coefficients can be extracted from the design matrix using the colnames() function.

SOLUTION

### make the posterior coefficient visualization for model 3
viz_post_coefs(laplace_03_weak$mode[-length(laplace_03_weak$mode)],
               sqrt(diag(laplace_03_weak$var_matrix))[-length(laplace_03_weak$mode)],
               info_03_weak$design_matrix %>% colnames())

### make the posterior coefficient visualization for model 6
viz_post_coefs(laplace_06_weak$mode[-length(laplace_06_weak$mode)],
               sqrt(diag(laplace_06_weak$var_matrix))[-length(laplace_06_weak$mode)],
               info_06_weak$design_matrix %>% colnames())

3e)

Use the Bayes Factor to identify the better of the models.

SOLUTION

### add more code chunks if you like
log_evidences <- c(laplace_03_weak$log_evidence, laplace_06_weak$log_evidence)
if (exp(log_evidences[1]) > exp(log_evidences[2])) {
  sprintf("Model 3 is better and has bayes factor of:%f", exp(log_evidences[1])/sum(exp(log_evidences)))
} else {
  sprintf("Model 6 is better and has bayes factor of:%f", exp(log_evidences[2])/sum(exp(log_evidences)))
}

## [1] "Model 3 is better and has bayes factor of:1.000000"

3f)

You fit the Bayesian models assuming a diffuse or weak prior. Let’s now try a more informative or strong prior by reducing the prior standard deviation on the regression coefficients from 50 to 1. The prior mean will still be zero.

Complete the first code chunk below, which defines the list of required information for both the model 3 and model 6 formulations using the strong prior on the regression coefficients. All other information, data and the \(\sigma\) prior, are the same as before.

Run the Laplace Approximation using the strong prior for both the model 3 and model 6 formulations. Assign the results to laplace_03_strong and laplace_06_strong.

Confirm that the optimizations converged for both laplace approximation results.

SOLUTION

Define the lists of required information for the strong prior.

info_03_strong <- list(
  yobs = df$y,
  design_matrix = model.matrix(y ~ (x1 + I(x1^2))*(x2 + I(x2^2)), data = df),
  mu_beta = 0,
  tau_beta = 1,
  sigma_rate = 1 
)

info_06_strong <- list(
  yobs = df$y,
  design_matrix = model.matrix(y ~ splines::ns(x1, df = 12) * (x2 +I(x2^2) + I(x2^3) + I(x2^4)), data = df),
  mu_beta = 0,
  tau_beta = 1,
  sigma_rate = 1 
)

Execute the Laplace Approximation.

### add more code chunks if you like
laplace_03_strong <- my_laplace(start_guess_03,
                                lm_logpost,
                                info_03_strong)

laplace_03_strong

## $mode
##  [1]  0.651631489  0.162914437 -0.152730088 -0.051739385 -0.548524155
##  [6]  0.118435267 -0.079153776  0.002640972  0.017130016 -0.398726740
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  0.0135459998 -2.873370e-04 -6.802265e-03  3.977485e-04 -5.671532e-03
##  [2,] -0.0002873370  7.716907e-03  1.696332e-04  7.896033e-04 -5.465947e-04
##  [3,] -0.0068022649  1.696332e-04  7.455503e-03  1.562543e-04  3.314492e-03
##  [4,]  0.0003977485  7.896033e-04  1.562543e-04  8.086078e-03  5.740636e-04
##  [5,] -0.0056715321 -5.465947e-04  3.314492e-03  5.740636e-04  5.582248e-03
##  [6,]  0.0023593465 -3.371931e-05  1.560391e-04  5.950152e-04 -1.881054e-03
##  [7,] -0.0016154477 -2.928150e-03  5.844267e-04 -1.063987e-03  1.486913e-03
##  [8,]  0.0012803561  1.764095e-04 -5.707103e-04 -2.435585e-03 -1.561461e-03
##  [9,]  0.0024573094  3.145161e-04 -3.430520e-03 -8.954372e-04 -2.355658e-03
## [10,] -0.0001336134 -1.536557e-05  7.457318e-05  2.363612e-06  7.767143e-05
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,]  2.359346e-03 -0.0016154477  0.0012803561  2.457309e-03 -1.336134e-04
##  [2,] -3.371931e-05 -0.0029281504  0.0001764095  3.145161e-04 -1.536557e-05
##  [3,]  1.560391e-04  0.0005844267 -0.0005707103 -3.430520e-03  7.457318e-05
##  [4,]  5.950152e-04 -0.0010639870 -0.0024355848 -8.954372e-04  2.363612e-06
##  [5,] -1.881054e-03  0.0014869130 -0.0015614607 -2.355658e-03  7.767143e-05
##  [6,]  1.197126e-02 -0.0044269068  0.0041668287 -2.733656e-03 -4.241310e-05
##  [7,] -4.426907e-03  0.0049132924 -0.0025030729  8.204208e-04  3.285110e-05
##  [8,]  4.166829e-03 -0.0025030729  0.0044113495 -4.354029e-04 -2.628410e-05
##  [9,] -2.733656e-03  0.0008204208 -0.0004354029  3.417039e-03 -3.183327e-05
## [10,] -4.241310e-05  0.0000328511 -0.0000262841 -3.183327e-05  5.001177e-03
## 
## $log_evidence
## [1] -130.025
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 112

laplace_06_strong <- my_laplace(start_guess_06,
                                lm_logpost,
                                info_06_strong)
laplace_06_strong

## $mode
##  [1]  0.3212233451 -0.0287175403 -0.1004156846  0.0563850943  0.7265367354
##  [6]  0.6241984921  0.1763432345  0.2157951086  0.3251316206 -0.2593082249
## [11]  0.8798067880  0.1410587249 -0.0375962861 -0.2760227845 -0.5765233744
## [16] -0.0787870208 -0.1935412001  0.4183144344 -0.1790317991 -0.1461579133
## [21]  0.3161817413  0.4519048156  0.7715572767 -0.4720424034 -0.6946505909
## [26]  0.9825140738 -0.0213411466 -0.0923919177  0.2747168212  0.4445480819
## [31] -0.1194984204  0.4518601321  0.1670873690  0.1368712616 -0.2049106757
## [36] -0.2347101877  0.1132589673  0.0004572422 -0.0504898864  0.3036721392
## [41] -0.0703075815 -0.1455324735  0.3850220230 -0.0871126242  0.0073550637
## [46] -0.3653392780  0.2061344063  0.6149856026 -0.0792086730  0.4087060682
## [51]  0.0046394853  0.0792319406  0.1491674989  0.1128291515  0.1628276525
## [56]  0.5346678717 -0.1516368992  0.2565965590  0.0342450451  0.1988417877
## [61] -0.0787726228  0.0736476716  0.1600793707  0.3558464821 -0.0149379291
## [66] -0.5995065976
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  7.388224e-02 -0.0600681013 -0.0700384724 -6.318657e-02 -0.0692514124
##  [2,] -6.006810e-02  0.3030111866 -0.0522872815  9.295836e-02  0.0421116557
##  [3,] -7.003847e-02 -0.0522872815  0.2342916330 -1.080294e-02  0.0966653317
##  [4,] -6.318657e-02  0.0929583563 -0.0108029361  2.575281e-01 -0.0326693234
##  [5,] -6.925141e-02  0.0421116557  0.0966653317 -3.266932e-02  0.2391828267
##  [6,] -6.188450e-02  0.0601917161  0.0460860172  9.917915e-02 -0.0256958634
##  [7,] -5.925645e-02  0.0471795192  0.0624238493  3.886433e-02  0.0824027667
##  [8,] -6.645340e-02  0.0579107866  0.0632416563  6.225290e-02  0.0576272145
##  [9,] -6.680050e-02  0.0549305810  0.0646731466  5.578354e-02  0.0664241818
## [10,] -6.275211e-02  0.0543957378  0.0610999891  5.587478e-02  0.0599830225
## [11,] -4.630043e-02  0.0781851973  0.0243305942  4.552325e-02  0.0412837316
## [12,] -1.202120e-01  0.0020207454  0.1647911510  8.962006e-02  0.1235539896
## [13,] -1.887969e-03  0.0606090479 -0.0249700718  1.286167e-02 -0.0015117337
## [14,] -1.550560e-03  0.0124370826  0.0009073550 -4.000785e-03  0.0041239914
## [15,] -2.782347e-02  0.0098213878  0.0149969212  1.459756e-02  0.0124783230
## [16,]  4.094768e-03 -0.0049919506 -0.0052256016 -2.711798e-05 -0.0057922466
## [17,] -1.194747e-03  0.0109292630  0.0048035247  6.511931e-03  0.0074201188
## [18,]  9.086165e-03 -0.0153093590  0.0108009577 -7.697452e-03 -0.0095016989
## [19,] -2.827811e-03  0.0005823317  0.0298071030 -2.971565e-02  0.0200130649
## [20,] -3.680784e-03 -0.0029753677 -0.0109434612 -2.754010e-02 -0.0114300334
## [21,]  9.905912e-04 -0.0172679139  0.0128424858 -2.229154e-02  0.0247058699
## [22,]  1.124637e-03 -0.0084113607 -0.0034435675  1.035762e-02 -0.0095620917
## [23,] -2.047138e-05 -0.0077619854  0.0024672678  2.490237e-03 -0.0001262920
## [24,]  2.638240e-03 -0.0102166776 -0.0035820355  6.224214e-03 -0.0108061411
## [25,]  1.425452e-03 -0.0095139101 -0.0019130625  9.606913e-04 -0.0040557053
## [26,]  2.767152e-03 -0.0089495712 -0.0002779557  1.353545e-03 -0.0015901109
## [27,]  1.155304e-02 -0.0197241838 -0.0013991342 -9.570576e-03 -0.0089008510
## [28,] -1.114963e-02  0.0229455662 -0.0043476084  2.410593e-02  0.0091313209
## [29,]  3.493702e-03 -0.0153776881  0.0094426847 -5.883419e-03  0.0003000915
## [30,]  4.593264e-03 -0.1619183584  0.0367043535 -9.229096e-03  0.0090283630
## [31,]  4.485249e-03  0.0230946573 -0.1014358021 -1.408500e-02  0.0067107654
## [32,]  2.937857e-03  0.0160789042 -0.0048570093 -1.012573e-01  0.0035761798
## [33,]  1.044540e-02 -0.0002613912 -0.0132938279  3.704902e-02 -0.1437520392
## [34,]  1.164653e-02 -0.0026999051 -0.0060279160 -2.680409e-03  0.0177468264
## [35,]  9.633165e-03 -0.0018545111  0.0016260669 -1.128567e-02  0.0106749561
## [36,]  8.234431e-03 -0.0012174804 -0.0031510469 -4.632151e-04 -0.0040580096
## [37,]  1.093229e-03  0.0009841793  0.0015899668 -4.684053e-04  0.0029220415
## [38,]  1.082593e-02 -0.0031613652 -0.0038847634 -3.791794e-03 -0.0033756462
## [39,]  1.431648e-02 -0.0065692970 -0.0125575695 -8.660879e-03 -0.0096651076
## [40,] -8.427393e-03  0.0037433393  0.0209079351  1.107838e-02  0.0153559020
## [41,] -7.156218e-03  0.0068308752  0.0026213862  6.339391e-03  0.0053641775
## [42,]  4.902306e-03 -0.0037615938  0.0069189911 -1.888200e-03 -0.0025439472
## [43,] -1.008890e-02  0.0014170970 -0.0060749219  6.190936e-03  0.0058901702
## [44,] -2.399836e-03 -0.0064290921  0.0126928159  1.846267e-02 -0.0027625487
## [45,] -7.663466e-04  0.0078585955 -0.0002287436  1.404492e-03  0.0090972442
## [46,] -5.395781e-03  0.0050031235  0.0029050688  7.664331e-03 -0.0042367890
## [47,] -2.030070e-03  0.0017629535  0.0060188433 -6.714261e-03  0.0142903740
## [48,] -9.234420e-03  0.0052732048  0.0090208998  3.281950e-03  0.0098011625
## [49,] -3.520233e-03  0.0010156129  0.0032129542 -4.242595e-04  0.0031102082
## [50,] -6.783795e-03 -0.0014609575  0.0047945284  2.922520e-03  0.0028333876
## [51,] -2.364585e-04 -0.0034573613 -0.0029224820  5.355254e-06 -0.0036535825
## [52,] -2.767184e-03 -0.0029089369  0.0084403319 -3.784862e-03  0.0055735956
## [53,]  3.821541e-03 -0.0039456156 -0.0052043988 -3.496736e-04 -0.0046627554
## [54,]  4.294554e-03 -0.0025510122 -0.0050286015 -8.162264e-03 -0.0087720505
## [55,]  6.441789e-03 -0.0060657289  0.0049224436 -1.788830e-03 -0.0181082718
## [56,]  1.337726e-02 -0.0239182572  0.0015898082 -5.216645e-03  0.0119449599
## [57,]  3.003508e-03 -0.0115227812 -0.0084596966 -8.145852e-03  0.0070702220
## [58,]  8.684737e-03 -0.0119222143 -0.0061553781 -1.740817e-02  0.0017581762
## [59,]  2.800952e-03 -0.0126099530 -0.0084590441 -2.571806e-03 -0.0231336044
## [60,]  9.762911e-03 -0.0143443970 -0.0097046036 -1.275550e-02 -0.0101778200
## [61,]  1.040481e-02 -0.0115929275 -0.0094981683 -1.184073e-02 -0.0101530516
## [62,]  5.727726e-03 -0.0093742710 -0.0065709680 -9.369958e-03 -0.0078859861
## [63,] -3.419063e-03 -0.0089434166  0.0062413979 -1.684408e-03  0.0019287352
## [64,]  2.096018e-02 -0.0085640365 -0.0323566992 -2.398399e-02 -0.0266084112
## [65,]  3.528548e-04 -0.0113283603  0.0067236209 -2.681536e-03  0.0012885121
## [66,]  1.158708e-03 -0.0001972937 -0.0015276575 -4.302898e-04 -0.0019512122
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,] -0.0618844995 -5.925645e-02 -0.0664533976 -0.0668004993 -0.0627521099
##  [2,]  0.0601917161  4.717952e-02  0.0579107866  0.0549305810  0.0543957378
##  [3,]  0.0460860172  6.242385e-02  0.0632416563  0.0646731466  0.0610999891
##  [4,]  0.0991791472  3.886433e-02  0.0622528959  0.0557835430  0.0558747751
##  [5,] -0.0256958634  8.240277e-02  0.0576272145  0.0664241818  0.0599830225
##  [6,]  0.2843430685 -2.680902e-02  0.0792400982  0.0485566828  0.0599659814
##  [7,] -0.0268090151  3.358141e-01 -0.0424126902  0.0935445981  0.0335976685
##  [8,]  0.0792400982 -4.241269e-02  0.2434552860 -0.0090303616  0.0925857382
##  [9,]  0.0485566828  9.354460e-02 -0.0090303616  0.2401918620 -0.0343100525
## [10,]  0.0599659814  3.359767e-02  0.0925857382 -0.0343100525  0.3010958475
## [11,]  0.0377478608  4.536824e-02  0.0266351882  0.0825084646 -0.0698418099
## [12,]  0.1023341103  1.045175e-01  0.1124274543  0.1162886379  0.0937926773
## [13,]  0.0054659942 -2.319152e-03  0.0107851012 -0.0178497444  0.0540533253
## [14,]  0.0053374422 -6.206362e-03  0.0108385393  0.0031806247  0.0089942874
## [15,]  0.0071109322  6.727562e-03  0.0050703507  0.0177403804  0.0024876655
## [16,] -0.0062792546  6.902896e-05 -0.0093113674 -0.0046721281 -0.0082702415
## [17,]  0.0084636506  8.482114e-03  0.0107490866  0.0044297128  0.0112035864
## [18,] -0.0111975494 -1.557268e-03 -0.0158541323 -0.0093686753 -0.0143549788
## [19,] -0.0120825013  1.158507e-02 -0.0085830587  0.0009282996 -0.0056279819
## [20,]  0.0057565980  4.739539e-03 -0.0020088324  0.0011866409 -0.0023776218
## [21,] -0.0105548402  8.035785e-03 -0.0127841342 -0.0009330031 -0.0078869284
## [22,]  0.0166623605 -9.152857e-03 -0.0005199630 -0.0046244561 -0.0062778312
## [23,]  0.0029364951 -2.477981e-02 -0.0165024593  0.0052104601 -0.0010532986
## [24,]  0.0053273644 -5.140995e-02  0.0730009407 -0.0152753918 -0.0125677258
## [25,] -0.0053449889  3.340059e-03 -0.0114210684  0.0311337425 -0.0125742491
## [26,] -0.0027140856  8.329672e-03 -0.0087221226 -0.0193556733  0.0987053449
## [27,] -0.0081739379 -9.087774e-03 -0.0032847085 -0.0162965625 -0.0088092767
## [28,]  0.0101569521  2.207111e-02  0.0052140419  0.0134730938 -0.0094599783
## [29,] -0.0021555264  2.679413e-03 -0.0047344491  0.0085630785 -0.0280914594
## [30,]  0.0029186075  6.944260e-03  0.0049704428  0.0001276787  0.0063957015
## [31,] -0.0037197323 -3.634526e-03  0.0060218490 -0.0012705412  0.0052388762
## [32,]  0.0042803388 -1.327527e-03  0.0058419516 -0.0004914935  0.0048047717
## [33,]  0.0215279863  2.371236e-03 -0.0039568080 -0.0019915947  0.0008126642
## [34,] -0.1384300581 -7.193283e-03  0.0196557429 -0.0135815109  0.0074335223
## [35,]  0.0024036094 -1.595216e-01  0.0311265614 -0.0093338021  0.0053277888
## [36,]  0.0031159629  2.193105e-02 -0.1574426795  0.0170328667  0.0001583417
## [37,] -0.0025220752  1.802153e-02 -0.0040405982 -0.1203510718  0.0137242344
## [38,]  0.0010058367 -2.453537e-03 -0.0036658425  0.0135640518 -0.1169015851
## [39,] -0.0088995547 -1.557394e-03 -0.0153604403  0.0074083248 -0.0240260288
## [40,]  0.0141431937  1.972769e-02  0.0122584054  0.0197116388  0.0012933115
## [41,]  0.0041506954  5.443443e-03  0.0049529477  0.0070002297 -0.0083788455
## [42,]  0.0023587695 -3.818267e-03  0.0048398081 -0.0023487351  0.0045658150
## [43,]  0.0107924819  3.118079e-03  0.0112551181  0.0091369153  0.0100107699
## [44,]  0.0029372379 -1.384123e-02  0.0108639823  0.0016206451  0.0057058802
## [45,]  0.0088585513  5.642243e-03  0.0061588453  0.0026977396  0.0074822064
## [46,] -0.0169021762  1.376486e-02  0.0111647400  0.0030303034  0.0073406435
## [47,]  0.0080231452  1.436297e-02  0.0016509927  0.0049870597  0.0029090000
## [48,]  0.0041298556  2.250090e-02 -0.0201310705 -0.0100192511  0.0162104859
## [49,]  0.0032938360 -1.633667e-03 -0.0186461991  0.0272996454 -0.0149281682
## [50,]  0.0024708362  1.563866e-03 -0.0016969886  0.0042295336  0.0032242131
## [51,] -0.0025859767 -5.435892e-03 -0.0014378770 -0.0028543752 -0.0231738328
## [52,]  0.0045710181 -9.245620e-04  0.0084097174  0.0038813864 -0.0031713167
## [53,] -0.0041730747  4.473532e-04 -0.0074911824 -0.0012972374 -0.0097531565
## [54,] -0.0096883385 -8.844492e-03 -0.0114954856 -0.0066476782 -0.0115796947
## [55,] -0.0075183641 -1.221425e-02 -0.0127563604 -0.0083237355 -0.0131201275
## [56,] -0.0201079366  1.903333e-03 -0.0183787069 -0.0119988180 -0.0140774929
## [57,] -0.0055876135 -2.028674e-02 -0.0077161968 -0.0073193020 -0.0113459168
## [58,] -0.0049297443  1.853126e-02 -0.0221427199 -0.0072827997 -0.0147321171
## [59,]  0.0118014677 -7.632775e-03 -0.0057538018 -0.0089842000 -0.0089623848
## [60,] -0.0148932648 -8.273551e-03  0.0203518922  0.0051733678 -0.0162953866
## [61,] -0.0110789158 -1.280121e-02  0.0151154185 -0.0149850240 -0.0028197219
## [62,] -0.0078355277 -9.636434e-03 -0.0086586454 -0.0013658672 -0.0074101978
## [63,]  0.0000880042 -3.335098e-04  0.0021410025 -0.0049521897  0.0182957630
## [64,] -0.0233362959 -2.830469e-02 -0.0263816842 -0.0254259202 -0.0111545846
## [65,]  0.0024200786 -3.141482e-03  0.0025386953 -0.0009411538  0.0163542002
## [66,] -0.0017377743 -1.939218e-03 -0.0005929875 -0.0016450490 -0.0003281460
##               [,11]        [,12]         [,13]         [,14]         [,15]
##  [1,] -0.0463004273 -0.120211972 -1.887969e-03 -0.0015505603 -0.0278234736
##  [2,]  0.0781851973  0.002020745  6.060905e-02  0.0124370826  0.0098213878
##  [3,]  0.0243305942  0.164791151 -2.497007e-02  0.0009073550  0.0149969212
##  [4,]  0.0455232499  0.089620058  1.286167e-02 -0.0040007854  0.0145975582
##  [5,]  0.0412837316  0.123553990 -1.511734e-03  0.0041239914  0.0124783230
##  [6,]  0.0377478608  0.102334110  5.465994e-03  0.0053374422  0.0071109322
##  [7,]  0.0453682382  0.104517452 -2.319152e-03 -0.0062063622  0.0067275618
##  [8,]  0.0266351882  0.112427454  1.078510e-02  0.0108385393  0.0050703507
##  [9,]  0.0825084646  0.116288638 -1.784974e-02  0.0031806247  0.0177403804
## [10,] -0.0698418099  0.093792677  5.405333e-02  0.0089942874  0.0024876655
## [11,]  0.2416663520  0.013249780 -8.640244e-02  0.0043420986  0.0219564846
## [12,]  0.0132497795  0.450201427  5.776854e-02 -0.0067404439  0.0046553333
## [13,] -0.0864024398  0.057768543  3.824981e-01 -0.0002534069 -0.0074295011
## [14,]  0.0043420986 -0.006740444 -2.534069e-04  0.0859650645 -0.0082750507
## [15,]  0.0219564846  0.004655333 -7.429501e-03 -0.0082750507  0.1267424465
## [16,] -0.0022426453 -0.006367061 -1.990814e-03 -0.0293028348  0.0113354128
## [17,] -0.0008721441  0.020087810  3.200616e-03  0.0113532290 -0.0486902397
## [18,] -0.0138946082  0.008574603 -9.760842e-03 -0.0723831118  0.0014941560
## [19,]  0.0032917534 -0.003097846  5.676784e-03 -0.0637735948  0.0209357955
## [20,] -0.0011316288  0.012780409 -1.207074e-03 -0.0558134148  0.0116897266
## [21,] -0.0038931350  0.008095171 -1.226051e-03 -0.0733342168  0.0077739136
## [22,] -0.0028266023  0.004869116  4.925614e-05 -0.0646531073  0.0049324663
## [23,] -0.0040310596  0.012084163  1.944954e-03 -0.0596174850 -0.0079446962
## [24,] -0.0025448441  0.003513660 -1.233001e-03 -0.0688709420  0.0051271832
## [25,] -0.0058487753  0.003100137  6.420611e-03 -0.0558239321  0.0157941997
## [26,] -0.0340052573 -0.007987502 -2.962672e-03 -0.0556029090 -0.0122762870
## [27,] -0.0309799420  0.018986677  3.613291e-02 -0.0184613091 -0.0195820650
## [28,]  0.0321253121 -0.003025064 -2.186467e-02 -0.1007316314 -0.0206003098
## [29,]  0.0374506698 -0.025204235 -1.108706e-01  0.0030964330 -0.0155605885
## [30,] -0.0053939864  0.015552799  1.294462e-03  0.0055627539 -0.0608850550
## [31,] -0.0029944965  0.002718078  6.279973e-03  0.0128118910 -0.0477656194
## [32,]  0.0010476937 -0.001102025  7.304740e-03  0.0074101521 -0.0402089258
## [33,] -0.0085934497  0.006152593  3.132205e-03  0.0072945526 -0.0678388840
## [34,] -0.0132798654  0.008578665  5.115408e-03  0.0023920351 -0.0788821200
## [35,] -0.0094172069  0.011658128  5.331548e-03 -0.0006092628 -0.0774130187
## [36,] -0.0078602878  0.005379584  4.408650e-03 -0.0036522306 -0.0548535820
## [37,]  0.0000804220  0.005644967  6.421238e-04  0.0035082619 -0.0228266312
## [38,] -0.0072144930  0.008585064  9.106021e-03 -0.0019547449 -0.0672227179
## [39,] -0.1172718790  0.027128367  2.998261e-02 -0.0017470552 -0.0379104819
## [40,]  0.0360261403 -0.112521214 -5.914378e-03 -0.0247463941 -0.0615965514
## [41,]  0.0256608120 -0.004501557 -1.973703e-01 -0.0036784659  0.0131160500
## [42,] -0.0138128212  0.022513240 -8.497264e-03  0.0229421640 -0.0377273136
## [43,]  0.0084578950  0.006553564  3.296117e-03  0.0246423003  0.0073635222
## [44,] -0.0018403275  0.010767822 -1.168568e-03  0.0148523490 -0.0103747110
## [45,]  0.0009136342  0.002129096  4.586533e-03  0.0278851731 -0.0240587188
## [46,]  0.0020160696  0.008019648  1.777476e-03  0.0146158745 -0.0027476912
## [47,]  0.0022566350  0.002129393  2.138568e-03  0.0180893930 -0.0072691130
## [48,]  0.0042759070  0.008960151 -6.988548e-04  0.0136571212  0.0073141001
## [49,]  0.0104506381  0.006850145  4.017773e-03  0.0098287613  0.0078064506
## [50,]  0.0066261156 -0.003787253 -1.423074e-02 -0.0139572045  0.0268347644
## [51,]  0.0017895006 -0.001090782  1.484821e-02 -0.0144453511  0.0205640979
## [52,]  0.0065478130  0.008358124 -2.077287e-02  0.0137335752 -0.0112652160
## [53,]  0.0166409563 -0.027009025 -4.007133e-02 -0.0203988764 -0.0024847277
## [54,] -0.0060094586 -0.010352882 -7.555793e-03 -0.0146406521  0.0357976537
## [55,] -0.0017450861 -0.026325346 -9.902677e-04 -0.0102226959  0.0324782594
## [56,] -0.0095887018 -0.021963409 -2.010189e-03 -0.0114688384  0.0018710024
## [57,] -0.0009120609 -0.021347410 -2.891795e-03 -0.0116544096  0.0440419804
## [58,] -0.0029671368 -0.026424028 -2.418905e-04 -0.0136186439  0.0221775652
## [59,] -0.0014373738 -0.019370235 -3.812127e-03 -0.0007215817  0.0415663433
## [60,] -0.0073458634 -0.025967848 -3.332973e-03 -0.0059462236  0.0155785247
## [61,] -0.0016953214 -0.014597513  6.413762e-03  0.0043906048  0.0003722837
## [62,]  0.0052562254 -0.016301979 -1.736189e-03  0.0072477418  0.0176506554
## [63,]  0.0159555744 -0.011284878 -1.017452e-02  0.0052925822  0.0190139896
## [64,] -0.0209816906 -0.030288274  9.937924e-03  0.0045919993  0.0347583997
## [65,] -0.0122245356  0.013875300  5.171820e-02  0.0104689558 -0.0104446211
## [66,] -0.0028224623 -0.001529064  1.338725e-03  0.0009945125  0.0006510796
##               [,16]         [,17]         [,18]         [,19]         [,20]
##  [1,]  4.094768e-03 -0.0011947474  0.0090861650 -0.0028278106 -0.0036807843
##  [2,] -4.991951e-03  0.0109292630 -0.0153093590  0.0005823317 -0.0029753677
##  [3,] -5.225602e-03  0.0048035247  0.0108009577  0.0298071030 -0.0109434612
##  [4,] -2.711798e-05  0.0065119310 -0.0076974524 -0.0297156511 -0.0275401021
##  [5,] -5.792247e-03  0.0074201188 -0.0095016989  0.0200130649 -0.0114300334
##  [6,] -6.279255e-03  0.0084636506 -0.0111975494 -0.0120825013  0.0057565980
##  [7,]  6.902896e-05  0.0084821137 -0.0015572680  0.0115850661  0.0047395391
##  [8,] -9.311367e-03  0.0107490866 -0.0158541323 -0.0085830587 -0.0020088324
##  [9,] -4.672128e-03  0.0044297128 -0.0093686753  0.0009282996  0.0011866409
## [10,] -8.270242e-03  0.0112035864 -0.0143549788 -0.0056279819 -0.0023776218
## [11,] -2.242645e-03 -0.0008721441 -0.0138946082  0.0032917534 -0.0011316288
## [12,] -6.367061e-03  0.0200878103  0.0085746031 -0.0030978463  0.0127804095
## [13,] -1.990814e-03  0.0032006160 -0.0097608419  0.0056767835 -0.0012070739
## [14,] -2.930283e-02  0.0113532290 -0.0723831118 -0.0637735948 -0.0558134148
## [15,]  1.133541e-02 -0.0486902397  0.0014941560  0.0209357955  0.0116897266
## [16,]  5.891249e-02  0.0170854336  0.0234830580  0.0157188459  0.0149857459
## [17,]  1.708543e-02  0.0512074770 -0.0144368136 -0.0178860112 -0.0119570177
## [18,]  2.348306e-02 -0.0144368136  0.4801862601 -0.0585122227  0.0654221228
## [19,]  1.571885e-02 -0.0178860112 -0.0585122227  0.4844140003 -0.0900967092
## [20,]  1.498575e-02 -0.0119570177  0.0654221228 -0.0900967092  0.5626672032
## [21,]  2.156564e-02 -0.0121960171  0.0602407700  0.0824856890 -0.0646324792
## [22,]  1.584630e-02 -0.0112439757  0.0560224146  0.0402287242  0.0695429368
## [23,]  1.465333e-02 -0.0038872126  0.0526407524  0.0412475653  0.0333480793
## [24,]  1.954358e-02 -0.0100302028  0.0572132270  0.0496034994  0.0493913216
## [25,]  1.495298e-02 -0.0142737667  0.0453391417  0.0447523524  0.0363183930
## [26,]  4.908647e-03 -0.0051034068  0.0496087928  0.0395018717  0.0360865973
## [27,]  3.704105e-03  0.0068874759  0.0720447858 -0.0024755876  0.0129930280
## [28,]  9.391763e-03 -0.0104148189 -0.0537091120  0.1098994418  0.0613607792
## [29,]  3.358917e-03  0.0099566978  0.0796134591 -0.0259503044 -0.0004122913
## [30,] -1.917001e-02  0.0157663620 -0.0385921907  0.0122655683 -0.0286546842
## [31,] -1.524097e-02  0.0131051233 -0.0115105902  0.1078438906  0.0001634335
## [32,] -1.465080e-02  0.0097740163 -0.0211475265  0.0138848702  0.0461764744
## [33,] -1.295229e-02  0.0215029134  0.0017360235 -0.0304481370 -0.0113240171
## [34,] -6.601193e-03  0.0288206682  0.0052309297 -0.0220584783 -0.0065160484
## [35,] -6.224033e-03  0.0289383738  0.0021149276 -0.0035059057 -0.0043587799
## [36,] -1.079981e-03  0.0187972793  0.0044882082 -0.0025086953  0.0021579337
## [37,] -4.485427e-03  0.0049005263 -0.0008620536 -0.0050592129 -0.0031700368
## [38,] -6.637146e-03  0.0215367064  0.0055728175 -0.0061989731 -0.0014844076
## [39,]  4.026890e-03  0.0180535934  0.0042851123  0.0004059684 -0.0018730783
## [40,] -2.458955e-02 -0.0035182081  0.0223313580  0.0050202770  0.0176300084
## [41,]  2.393814e-03  0.0011181282  0.0028572920  0.0101625571  0.0024071431
## [42,] -3.928372e-02  0.0042853457 -0.1881766846 -0.0129195896 -0.0074190766
## [43,] -6.105479e-02 -0.0271053744  0.0178245506 -0.1057697631  0.0350454501
## [44,] -4.937874e-02 -0.0172296043 -0.0066161324  0.0061481908 -0.1990578033
## [45,] -5.836645e-02 -0.0110431312 -0.0264523774 -0.0094468771  0.0268459632
## [46,] -4.653032e-02 -0.0176755177 -0.0108442432 -0.0119473136  0.0046056483
## [47,] -5.358833e-02 -0.0200814415 -0.0175144129  0.0055311480 -0.0219545646
## [48,] -5.162047e-02 -0.0258775896 -0.0100333134 -0.0024332523 -0.0037737394
## [49,] -3.559652e-02 -0.0211591601 -0.0076982738 -0.0004583106 -0.0039689096
## [50,] -2.825869e-02 -0.0340259211  0.0132704723  0.0182073970  0.0138555225
## [51,] -6.047327e-04 -0.0258182183 -0.0107949434  0.0171422706  0.0117313529
## [52,] -8.375854e-02 -0.0218082123  0.0484875770 -0.0059452985 -0.0037032315
## [53,]  2.049140e-02 -0.0104309434 -0.0138010942  0.0161595856  0.0126824673
## [54,] -1.020901e-02 -0.0374301127 -0.0211102545  0.0075658279  0.0196640620
## [55,] -1.795302e-02 -0.0457783057  0.0253386903 -0.0088998758  0.0168603364
## [56,] -1.491377e-02 -0.0287688242  0.0165779206  0.0220283358 -0.0315615880
## [57,] -1.680990e-02 -0.0495195041  0.0162950299  0.0130501084  0.0207189220
## [58,] -1.390938e-02 -0.0372089710  0.0099525890  0.0341609078  0.0195169423
## [59,] -2.495793e-02 -0.0501930591  0.0117159623 -0.0049922873  0.0013617384
## [60,] -2.232423e-02 -0.0379921136  0.0112608763  0.0106829226  0.0062238459
## [61,] -2.694170e-02 -0.0275582711  0.0015565738  0.0006178859 -0.0005272086
## [62,] -3.115618e-02 -0.0376627339 -0.0012103870  0.0017049453 -0.0010980369
## [63,] -2.384846e-02 -0.0224723303 -0.0086415036 -0.0023993601 -0.0000041520
## [64,] -5.142753e-02 -0.0852799873  0.0248867928  0.0185190816  0.0022506025
## [65,] -3.254522e-02 -0.0057662276 -0.0169423414 -0.0104708540 -0.0038647662
## [66,]  1.170030e-03  0.0007830403 -0.0033507079  0.0004893545 -0.0001856978
##               [,21]         [,22]         [,23]         [,24]         [,25]
##  [1,]  0.0009905912  1.124637e-03 -2.047138e-05  0.0026382395  0.0014254522
##  [2,] -0.0172679139 -8.411361e-03 -7.761985e-03 -0.0102166776 -0.0095139101
##  [3,]  0.0128424858 -3.443568e-03  2.467268e-03 -0.0035820355 -0.0019130625
##  [4,] -0.0222915417  1.035762e-02  2.490237e-03  0.0062242137  0.0009606913
##  [5,]  0.0247058699 -9.562092e-03 -1.262920e-04 -0.0108061411 -0.0040557053
##  [6,] -0.0105548402  1.666236e-02  2.936495e-03  0.0053273644 -0.0053449889
##  [7,]  0.0080357848 -9.152857e-03 -2.477981e-02 -0.0514099480  0.0033400592
##  [8,] -0.0127841342 -5.199630e-04 -1.650246e-02  0.0730009407 -0.0114210684
##  [9,] -0.0009330031 -4.624456e-03  5.210460e-03 -0.0152753918  0.0311337425
## [10,] -0.0078869284 -6.277831e-03 -1.053299e-03 -0.0125677258 -0.0125742491
## [11,] -0.0038931350 -2.826602e-03 -4.031060e-03 -0.0025448441 -0.0058487753
## [12,]  0.0080951712  4.869116e-03  1.208416e-02  0.0035136597  0.0031001372
## [13,] -0.0012260508  4.925614e-05  1.944954e-03 -0.0012330011  0.0064206111
## [14,] -0.0733342168 -6.465311e-02 -5.961749e-02 -0.0688709420 -0.0558239321
## [15,]  0.0077739136  4.932466e-03 -7.944696e-03  0.0051271832  0.0157941997
## [16,]  0.0215656378  1.584630e-02  1.465333e-02  0.0195435832  0.0149529833
## [17,] -0.0121960171 -1.124398e-02 -3.887213e-03 -0.0100302028 -0.0142737667
## [18,]  0.0602407700  5.602241e-02  5.264075e-02  0.0572132270  0.0453391417
## [19,]  0.0824856890  4.022872e-02  4.124757e-02  0.0496034994  0.0447523524
## [20,] -0.0646324792  6.954294e-02  3.334808e-02  0.0493913216  0.0363183930
## [21,]  0.3985634647 -7.230120e-02  8.053227e-02  0.0439775496  0.0515345259
## [22,] -0.0723012018  4.814463e-01 -8.584341e-02  0.0732963339  0.0361663814
## [23,]  0.0805322729 -8.584341e-02  4.941761e-01 -0.0582460239  0.0689971979
## [24,]  0.0439775496  7.329633e-02 -5.824602e-02  0.4048288548 -0.0618108665
## [25,]  0.0515345259  3.616638e-02  6.899720e-02 -0.0618108665  0.4876428254
## [26,]  0.0479597816  4.708737e-02  3.269186e-02  0.0597292754 -0.0732938569
## [27,]  0.0159775934  1.250564e-02  1.988506e-02  0.0091625149  0.0526224773
## [28,]  0.0871867164  7.778509e-02  7.688977e-02  0.0809293817  0.0796569739
## [29,] -0.0024460183 -2.387936e-03 -4.610441e-04 -0.0024943591 -0.0093618099
## [30,]  0.0016892519 -3.303991e-03  4.028920e-03 -0.0052882224 -0.0094146334
## [31,] -0.0101833530 -8.449178e-03  2.852316e-03 -0.0080118565 -0.0108513670
## [32,] -0.0222847109 -7.446789e-03  3.933102e-03 -0.0047236307 -0.0088418078
## [33,]  0.0103195083  1.435111e-02  1.362682e-02 -0.0069728316 -0.0106288516
## [34,] -0.0009929918  1.453538e-02 -1.947406e-02  0.0130949342 -0.0084622610
## [35,]  0.0042632502 -2.543793e-02 -2.686682e-02 -0.0039982020 -0.0052469569
## [36,] -0.0037023467  1.963643e-03 -2.724648e-02  0.0068838175  0.0086986000
## [37,] -0.0029486063 -5.136123e-04 -1.138956e-02 -0.0049065696  0.0757103779
## [38,]  0.0012264302  3.225537e-03  1.244339e-02  0.0017668467 -0.0029284061
## [39,]  0.0003825248  9.226432e-04  1.406026e-03  0.0045086459 -0.0056230989
## [40,]  0.0235028581  2.293748e-02  2.740843e-02  0.0217168675  0.0129583713
## [41,]  0.0031804225  1.904314e-03  2.249450e-03  0.0023584972  0.0082144734
## [42,] -0.0203083083 -1.237865e-02 -9.964098e-03 -0.0139905382 -0.0136767459
## [43,] -0.0224161773 -1.151182e-02 -1.288178e-02 -0.0167454048 -0.0103495015
## [44,]  0.0144628968 -6.016344e-03  6.602750e-03 -0.0032187651 -0.0069998486
## [45,] -0.1161079657  2.394880e-02 -2.986009e-02 -0.0198359819 -0.0146234040
## [46,] -0.0246733530 -2.075277e-01  5.568205e-02  0.0165340797 -0.0125798747
## [47,]  0.0299054286  4.182268e-02 -2.160156e-01  0.0041046787 -0.0096217584
## [48,] -0.0089346925 -8.080244e-03  2.013930e-03 -0.1933558282  0.0349063248
## [49,] -0.0014179692 -1.289954e-02  3.876961e-02  0.0272866921 -0.1724471170
## [50,]  0.0136589692  1.669820e-02  4.506235e-03  0.0170937021  0.0073643298
## [51,]  0.0128686433  1.056399e-02  1.133465e-02  0.0072908316  0.0412337934
## [52,] -0.0054387827 -9.103506e-04  1.046679e-03 -0.0059426914  0.0071490221
## [53,]  0.0164750373  1.398667e-02  1.228483e-02  0.0153814879  0.0066927125
## [54,]  0.0129779085  1.331876e-02  6.685595e-03  0.0131984076  0.0152347729
## [55,]  0.0091313902  1.123348e-02  5.313678e-03  0.0091110676  0.0120273031
## [56,]  0.0219066469  8.841311e-03 -2.892239e-03  0.0090327129  0.0104888489
## [57,] -0.0021539325  1.372027e-02  6.852178e-03  0.0111459749  0.0145347151
## [58,] -0.0065761205 -4.321186e-02  1.207730e-02  0.0191884403  0.0095007730
## [59,]  0.0093399196  6.564501e-02  3.978934e-02 -0.0006987365  0.0107786158
## [60,]  0.0095025959  2.585564e-03  3.146309e-02  0.0372453554 -0.0125694862
## [61,] -0.0021165105  1.478529e-03 -1.982739e-02 -0.0089135226  0.0305695904
## [62,] -0.0033454646 -2.519627e-03 -3.437524e-03 -0.0057299641  0.0120083138
## [63,] -0.0021546259 -1.026854e-03 -4.365973e-03 -0.0016010097 -0.0139470508
## [64,]  0.0007934607  3.639385e-03 -5.864836e-03  0.0006101185 -0.0033851188
## [65,] -0.0076409833 -3.111484e-03 -5.038822e-03 -0.0043061557 -0.0116242894
## [66,] -0.0020667251 -2.289379e-03 -3.596608e-03  0.0017223553  0.0024737492
##               [,26]         [,27]         [,28]         [,29]         [,30]
##  [1,]  0.0027671523  0.0115530410 -0.0111496329  0.0034937018  0.0045932640
##  [2,] -0.0089495712 -0.0197241838  0.0229455662 -0.0153776881 -0.1619183584
##  [3,] -0.0002779557 -0.0013991342 -0.0043476084  0.0094426847  0.0367043535
##  [4,]  0.0013535450 -0.0095705757  0.0241059290 -0.0058834187 -0.0092290957
##  [5,] -0.0015901109 -0.0089008510  0.0091313209  0.0003000915  0.0090283630
##  [6,] -0.0027140856 -0.0081739379  0.0101569521 -0.0021555264  0.0029186075
##  [7,]  0.0083296722 -0.0090877745  0.0220711068  0.0026794133  0.0069442604
##  [8,] -0.0087221226 -0.0032847085  0.0052140419 -0.0047344491  0.0049704428
##  [9,] -0.0193556733 -0.0162965625  0.0134730938  0.0085630785  0.0001276787
## [10,]  0.0987053449 -0.0088092767 -0.0094599783 -0.0280914594  0.0063957015
## [11,] -0.0340052573 -0.0309799420  0.0321253121  0.0374506698 -0.0053939864
## [12,] -0.0079875021  0.0189866770 -0.0030250635 -0.0252042353  0.0155527991
## [13,] -0.0029626717  0.0361329141 -0.0218646687 -0.1108706366  0.0012944623
## [14,] -0.0556029090 -0.0184613091 -0.1007316314  0.0030964330  0.0055627539
## [15,] -0.0122762870 -0.0195820650 -0.0206003098 -0.0155605885 -0.0608850550
## [16,]  0.0049086466  0.0037041054  0.0093917628  0.0033589174 -0.0191700091
## [17,] -0.0051034068  0.0068874759 -0.0104148189  0.0099566978  0.0157663620
## [18,]  0.0496087928  0.0720447858 -0.0537091120  0.0796134591 -0.0385921907
## [19,]  0.0395018717 -0.0024755876  0.1098994418 -0.0259503044  0.0122655683
## [20,]  0.0360865973  0.0129930280  0.0613607792 -0.0004122913 -0.0286546842
## [21,]  0.0479597816  0.0159775934  0.0871867164 -0.0024460183  0.0016892519
## [22,]  0.0470873668  0.0125056375  0.0777850880 -0.0023879361 -0.0033039911
## [23,]  0.0326918648  0.0198850648  0.0768897663 -0.0004610441  0.0040289204
## [24,]  0.0597292754  0.0091625149  0.0809293817 -0.0024943591 -0.0052882224
## [25,] -0.0732938569  0.0526224773  0.0796569739 -0.0093618099 -0.0094146334
## [26,]  0.5980064558 -0.1487498475  0.0124006753  0.0102321875  0.0112843160
## [27,] -0.1487498475  0.5382050953  0.0350987734 -0.0517660899 -0.0044689883
## [28,]  0.0124006753  0.0350987734  0.6082627308  0.0505171977  0.0420194116
## [29,]  0.0102321875 -0.0517660899  0.0505171977  0.5641712011 -0.0085168668
## [30,]  0.0112843160 -0.0044689883  0.0420194116 -0.0085168668  0.6906966151
## [31,] -0.0017490551  0.0045191935  0.0031585707  0.0036598110 -0.0414641099
## [32,]  0.0051155087  0.0008423422  0.0122662212  0.0009759210  0.0355431223
## [33,]  0.0060365596  0.0096919001  0.0103025519  0.0088824646  0.0209051733
## [34,]  0.0082587029  0.0130348191  0.0160754834  0.0089668389  0.0311246598
## [35,]  0.0104000167  0.0139339890  0.0235674636  0.0097594118  0.0396415958
## [36,]  0.0022371971  0.0117897750  0.0195348433  0.0049536192  0.0270032793
## [37,] -0.0002275872  0.0014975032  0.0062569335  0.0082401365  0.0124479634
## [38,]  0.0449843932 -0.0358092927 -0.0033908968 -0.0004046461  0.0345794980
## [39,] -0.0388373216 -0.0911692265  0.0075892470  0.0479816929  0.0453716606
## [40,]  0.0074698441  0.0033291503 -0.0109491633 -0.0517410239 -0.0319446867
## [41,] -0.0116172940  0.0281644702 -0.0588823014 -0.1644537928  0.0360479393
## [42,] -0.0040655100 -0.0145387940  0.0391825973 -0.0243600305 -0.1611309530
## [43,] -0.0023071203 -0.0025461081 -0.0149106121 -0.0006269794  0.0723154405
## [44,]  0.0028607637  0.0008566042  0.0025096656 -0.0017683566  0.0060364863
## [45,] -0.0023139799 -0.0020187846 -0.0031361669 -0.0030200354  0.0328497116
## [46,]  0.0022903616 -0.0021515663  0.0025703770 -0.0046692456  0.0094758323
## [47,]  0.0002723998 -0.0010527307  0.0041816978 -0.0038088073  0.0228730092
## [48,]  0.0245697281 -0.0160021861 -0.0010638109 -0.0061548396  0.0116465968
## [49,] -0.0331378364  0.0278011417  0.0208304321  0.0061745147  0.0054855285
## [50,] -0.2028860386 -0.0780759110  0.0041674763  0.0092417866 -0.0026143970
## [51,] -0.0639551113 -0.1954143580  0.0284954966  0.0287174174 -0.0101121493
## [52,] -0.0070594573 -0.0030352531 -0.1437869877  0.0183116814  0.0297959804
## [53,]  0.0195503102  0.0028102983  0.0298814881 -0.1587028970 -0.0031868674
## [54,]  0.0069622195 -0.0008744469  0.0104048152 -0.0053935949 -0.1639790787
## [55,]  0.0067169958 -0.0044089100  0.0118246155 -0.0075879118 -0.0030354815
## [56,]  0.0103639016  0.0018062115  0.0180972314 -0.0032963732  0.0136204137
## [57,]  0.0054394425 -0.0055588092  0.0111261407 -0.0092675959 -0.0169333533
## [58,]  0.0107573567 -0.0035114542  0.0196117530 -0.0077917616  0.0102692145
## [59,] -0.0014292657 -0.0055743006 -0.0003622190 -0.0073857574 -0.0225625949
## [60,]  0.0111757557 -0.0118011128  0.0036729331 -0.0130435515  0.0007983492
## [61,] -0.0130703166  0.0293780376  0.0284116767  0.0161001550  0.0063774196
## [62,]  0.0318301289  0.0575276360  0.0157411733 -0.0217201417  0.0001260029
## [63,]  0.0582830426  0.0515255763  0.0037408808 -0.0203009885 -0.0162730283
## [64,]  0.0384262139 -0.0030165825  0.0263094176 -0.0169106353  0.0355395545
## [65,]  0.0251594869 -0.0111199347 -0.0001743791  0.0243429041 -0.0082050518
## [66,] -0.0062964122  0.0022259379 -0.0006461750 -0.0020588684 -0.0041371575
##               [,31]         [,32]         [,33]         [,34]         [,35]
##  [1,]  0.0044852495  0.0029378567  1.044540e-02  0.0116465260  0.0096331652
##  [2,]  0.0230946573  0.0160789042 -2.613912e-04 -0.0026999051 -0.0018545111
##  [3,] -0.1014358021 -0.0048570093 -1.329383e-02 -0.0060279160  0.0016260669
##  [4,] -0.0140850045 -0.1012572859  3.704902e-02 -0.0026804089 -0.0112856736
##  [5,]  0.0067107654  0.0035761798 -1.437520e-01  0.0177468264  0.0106749561
##  [6,] -0.0037197323  0.0042803388  2.152799e-02 -0.1384300581  0.0024036094
##  [7,] -0.0036345263 -0.0013275269  2.371236e-03 -0.0071932826 -0.1595215824
##  [8,]  0.0060218490  0.0058419516 -3.956808e-03  0.0196557429  0.0311265614
##  [9,] -0.0012705412 -0.0004914935 -1.991595e-03 -0.0135815109 -0.0093338021
## [10,]  0.0052388762  0.0048047717  8.126642e-04  0.0074335223  0.0053277888
## [11,] -0.0029944965  0.0010476937 -8.593450e-03 -0.0132798654 -0.0094172069
## [12,]  0.0027180780 -0.0011020247  6.152593e-03  0.0085786648  0.0116581279
## [13,]  0.0062799729  0.0073047397  3.132205e-03  0.0051154079  0.0053315483
## [14,]  0.0128118910  0.0074101521  7.294553e-03  0.0023920351 -0.0006092628
## [15,] -0.0477656194 -0.0402089258 -6.783888e-02 -0.0788821200 -0.0774130187
## [16,] -0.0152409724 -0.0146508038 -1.295229e-02 -0.0066011932 -0.0062240326
## [17,]  0.0131051233  0.0097740163  2.150291e-02  0.0288206682  0.0289383738
## [18,] -0.0115105902 -0.0211475265  1.736023e-03  0.0052309297  0.0021149276
## [19,]  0.1078438906  0.0138848702 -3.044814e-02 -0.0220584783 -0.0035059057
## [20,]  0.0001634335  0.0461764744 -1.132402e-02 -0.0065160484 -0.0043587799
## [21,] -0.0101833530 -0.0222847109  1.031951e-02 -0.0009929918  0.0042632502
## [22,] -0.0084491782 -0.0074467892  1.435111e-02  0.0145353815 -0.0254379256
## [23,]  0.0028523165  0.0039331016  1.362682e-02 -0.0194740584 -0.0268668207
## [24,] -0.0080118565 -0.0047236307 -6.972832e-03  0.0130949342 -0.0039982020
## [25,] -0.0108513670 -0.0088418078 -1.062885e-02 -0.0084622610 -0.0052469569
## [26,] -0.0017490551  0.0051155087  6.036560e-03  0.0082587029  0.0104000167
## [27,]  0.0045191935  0.0008423422  9.691900e-03  0.0130348191  0.0139339890
## [28,]  0.0031585707  0.0122662212  1.030255e-02  0.0160754834  0.0235674636
## [29,]  0.0036598110  0.0009759210  8.882465e-03  0.0089668389  0.0097594118
## [30,] -0.0414641099  0.0355431223  2.090517e-02  0.0311246598  0.0396415958
## [31,]  0.7066082287 -0.0816284315  6.195127e-02  0.0495229334  0.0268662957
## [32,] -0.0816284315  0.7411256240 -2.839520e-02  0.0479609582  0.0190539895
## [33,]  0.0619512658 -0.0283951959  5.885036e-01 -0.1015380994  0.0761686037
## [34,]  0.0495229334  0.0479609582 -1.015381e-01  0.6023047719 -0.0929272157
## [35,]  0.0268662957  0.0190539895  7.616860e-02 -0.0929272157  0.5725782740
## [36,]  0.0167808631  0.0162055873  2.034655e-02  0.0402628474 -0.0596552287
## [37,]  0.0109083227  0.0085914116  1.651560e-02  0.0105088851  0.0248368621
## [38,]  0.0251231317  0.0225759274  3.515803e-02  0.0454025980  0.0348757503
## [39,]  0.0139750633  0.0089584441  2.017755e-02  0.0223044466  0.0230225770
## [40,]  0.0252046636  0.0285059375  3.782276e-02  0.0415819264  0.0444258622
## [41,] -0.0037277015 -0.0058301582 -6.471190e-03 -0.0087247491 -0.0060194242
## [42,] -0.0333539952  0.0175480637  2.787745e-02  0.0255081700  0.0219885936
## [43,]  0.0574855530  0.0387726829  3.999650e-03 -0.0088778925 -0.0044329648
## [44,]  0.0397257769 -0.1000288108  6.025124e-02  0.0582121923 -0.0045909862
## [45,] -0.0002090901  0.0371083318 -5.414617e-02 -0.0091269477  0.0164780310
## [46,]  0.0134744423  0.0146170839  1.517312e-02 -0.0044486096  0.0221294377
## [47,] -0.0027339978  0.0059866825 -3.066608e-02  0.0433685783  0.0337413271
## [48,]  0.0054142827  0.0087015150  2.841669e-03 -0.0119017732 -0.0037170661
## [49,]  0.0032139623  0.0040144802 -1.301667e-03 -0.0042483479 -0.0150228002
## [50,] -0.0057107139 -0.0015653016 -1.067413e-02 -0.0140424660 -0.0188413031
## [51,] -0.0108468271 -0.0054213620 -9.145045e-03 -0.0099459320 -0.0113955434
## [52,]  0.0285635489  0.0179415407  1.355674e-02  0.0052048370  0.0066785728
## [53,] -0.0082865445 -0.0008501486  1.811041e-03  0.0036598405  0.0041947514
## [54,] -0.0013752373 -0.0151558549 -1.561341e-02 -0.0204977801 -0.0211956602
## [55,] -0.1724997665  0.0391262767 -7.190498e-03 -0.0177993173 -0.0202134958
## [56,] -0.0040417009 -0.2427388081 -5.343361e-02 -0.0369788535  0.0096030609
## [57,] -0.0019874327  0.0204718055 -1.302140e-01  0.0344121640 -0.0426980726
## [58,] -0.0451495758 -0.0180596044 -3.157180e-02 -0.1937802574  0.0435557977
## [59,]  0.0227605664  0.0027099528  3.468144e-02  0.0158351936 -0.2003819682
## [60,] -0.0010815190  0.0007178985 -4.128837e-03 -0.0092962666  0.0162174918
## [61,]  0.0074815475  0.0057906434  7.481058e-03 -0.0041560291  0.0176574177
## [62,]  0.0012658243  0.0014278514 -3.785176e-03 -0.0105085502 -0.0102769940
## [63,] -0.0067502962 -0.0006726951 -6.847849e-03 -0.0129388328 -0.0094127367
## [64,]  0.0114441878  0.0035852475 -6.747377e-03 -0.0181531198 -0.0229138158
## [65,]  0.0023271649  0.0084658492  6.995576e-03  0.0074794065  0.0018988666
## [66,]  0.0012970365 -0.0026370886 -8.338188e-05  0.0002255103  0.0007055726
##               [,36]         [,37]         [,38]         [,39]        [,40]
##  [1,]  0.0082344306  0.0010932290  0.0108259251  1.431648e-02 -0.008427393
##  [2,] -0.0012174804  0.0009841793 -0.0031613652 -6.569297e-03  0.003743339
##  [3,] -0.0031510469  0.0015899668 -0.0038847634 -1.255757e-02  0.020907935
##  [4,] -0.0004632151 -0.0004684053 -0.0037917941 -8.660879e-03  0.011078380
##  [5,] -0.0040580096  0.0029220415 -0.0033756462 -9.665108e-03  0.015355902
##  [6,]  0.0031159629 -0.0025220752  0.0010058367 -8.899555e-03  0.014143194
##  [7,]  0.0219310514  0.0180215271 -0.0024535370 -1.557394e-03  0.019727688
##  [8,] -0.1574426795 -0.0040405982 -0.0036658425 -1.536044e-02  0.012258405
##  [9,]  0.0170328667 -0.1203510718  0.0135640518  7.408325e-03  0.019711639
## [10,]  0.0001583417  0.0137242344 -0.1169015851 -2.402603e-02  0.001293312
## [11,] -0.0078602878  0.0000804220 -0.0072144930 -1.172719e-01  0.036026140
## [12,]  0.0053795837  0.0056449669  0.0085850645  2.712837e-02 -0.112521214
## [13,]  0.0044086497  0.0006421238  0.0091060211  2.998261e-02 -0.005914378
## [14,] -0.0036522306  0.0035082619 -0.0019547449 -1.747055e-03 -0.024746394
## [15,] -0.0548535820 -0.0228266312 -0.0672227179 -3.791048e-02 -0.061596551
## [16,] -0.0010799814 -0.0044854273 -0.0066371458  4.026890e-03 -0.024589548
## [17,]  0.0187972793  0.0049005263  0.0215367064  1.805359e-02 -0.003518208
## [18,]  0.0044882082 -0.0008620536  0.0055728175  4.285112e-03  0.022331358
## [19,] -0.0025086953 -0.0050592129 -0.0061989731  4.059684e-04  0.005020277
## [20,]  0.0021579337 -0.0031700368 -0.0014844076 -1.873078e-03  0.017630008
## [21,] -0.0037023467 -0.0029486063  0.0012264302  3.825248e-04  0.023502858
## [22,]  0.0019636429 -0.0005136123  0.0032255369  9.226432e-04  0.022937482
## [23,] -0.0272464795 -0.0113895614  0.0124433909  1.406026e-03  0.027408433
## [24,]  0.0068838175 -0.0049065696  0.0017668467  4.508646e-03  0.021716868
## [25,]  0.0086986000  0.0757103779 -0.0029284061 -5.623099e-03  0.012958371
## [26,]  0.0022371971 -0.0002275872  0.0449843932 -3.883732e-02  0.007469844
## [27,]  0.0117897750  0.0014975032 -0.0358092927 -9.116923e-02  0.003329150
## [28,]  0.0195348433  0.0062569335 -0.0033908968  7.589247e-03 -0.010949163
## [29,]  0.0049536192  0.0082401365 -0.0004046461  4.798169e-02 -0.051741024
## [30,]  0.0270032793  0.0124479634  0.0345794980  4.537166e-02 -0.031944687
## [31,]  0.0167808631  0.0109083227  0.0251231317  1.397506e-02  0.025204664
## [32,]  0.0162055873  0.0085914116  0.0225759274  8.958444e-03  0.028505938
## [33,]  0.0203465452  0.0165156045  0.0351580332  2.017755e-02  0.037822756
## [34,]  0.0402628474  0.0105088851  0.0454025980  2.230445e-02  0.041581926
## [35,] -0.0596552287  0.0248368621  0.0348757503  2.302258e-02  0.044425862
## [36,]  0.6181074143 -0.0348577212  0.0341873646  1.622260e-02  0.031738009
## [37,] -0.0348577212  0.7835551887 -0.0503542213  1.772782e-02  0.023497426
## [38,]  0.0341873646 -0.0503542213  0.7069738731 -1.642897e-01 -0.025376765
## [39,]  0.0162226050  0.0177278243 -0.1642897202  6.875261e-01 -0.002922829
## [40,]  0.0317380087  0.0234974257 -0.0253767655 -2.922829e-03  0.815610078
## [41,] -0.0062771984  0.0008766779  0.0093039058  4.370918e-03  0.031578201
## [42,]  0.0137414783  0.0076610747  0.0191867878  3.724131e-03  0.035165454
## [43,] -0.0069243786  0.0013412928 -0.0023982167 -1.041398e-02  0.020414828
## [44,] -0.0043478259  0.0050785208  0.0061067638 -2.619909e-03  0.025987982
## [45,]  0.0110544997  0.0055289013  0.0139532744 -1.282833e-04  0.030327593
## [46,]  0.0209728297  0.0016599873  0.0030050468 -4.218342e-03  0.020372994
## [47,] -0.0064659771  0.0078639925  0.0010369956 -4.223460e-03  0.026530687
## [48,]  0.0853664887  0.0294693894 -0.0043965955 -5.787793e-03  0.020784777
## [49,]  0.0251294321  0.0582375776 -0.0410975021 -1.487081e-02  0.022175879
## [50,] -0.0124879055  0.0110590561  0.0365055863  5.913616e-02  0.035949314
## [51,] -0.0030461208 -0.0122153154  0.0329522041 -2.356010e-02  0.009704547
## [52,] -0.0008406137 -0.0003688363  0.0205736754 -1.472481e-02 -0.003823182
## [53,]  0.0056692349  0.0070275229 -0.0002721699  4.833602e-05 -0.062332301
## [54,] -0.0134369816 -0.0042793422 -0.0160149979 -2.871770e-02  0.042041252
## [55,] -0.0123020375 -0.0018783211 -0.0128849221 -1.116027e-02  0.006323155
## [56,]  0.0092025187  0.0015176594  0.0034582847 -3.437152e-03  0.023088574
## [57,] -0.0194340680 -0.0043876149 -0.0191290900 -1.652244e-02  0.005078087
## [58,]  0.0110094230 -0.0012328970 -0.0072104379 -9.111820e-03  0.014349467
## [59,]  0.0004513667 -0.0022332963 -0.0219185454 -1.719229e-02  0.005505614
## [60,] -0.2433983317 -0.0324212208  0.0426194395 -9.412533e-03  0.011206521
## [61,] -0.0089762188 -0.1877971855 -0.1145288190 -6.328730e-03  0.037068878
## [62,] -0.0058876473 -0.0021033168 -0.1705208566  4.878245e-03  0.016027251
## [63,] -0.0111654676  0.0085965603  0.0154143475 -1.457908e-01  0.045573525
## [64,] -0.0121429037 -0.0010731077  0.0059237874 -7.912885e-03 -0.190474279
## [65,]  0.0030129565 -0.0125180928  0.0168483703 -7.011704e-02  0.049606744
## [66,]  0.0014262393 -0.0005401815 -0.0016109073  1.325791e-03 -0.003238743
##               [,41]         [,42]         [,43]         [,44]         [,45]
##  [1,] -0.0071562177  0.0049023061 -0.0100889045 -0.0023998360 -0.0007663466
##  [2,]  0.0068308752 -0.0037615938  0.0014170970 -0.0064290921  0.0078585955
##  [3,]  0.0026213862  0.0069189911 -0.0060749219  0.0126928159 -0.0002287436
##  [4,]  0.0063393912 -0.0018881997  0.0061909356  0.0184626655  0.0014044917
##  [5,]  0.0053641775 -0.0025439472  0.0058901702 -0.0027625487  0.0090972442
##  [6,]  0.0041506954  0.0023587695  0.0107924819  0.0029372379  0.0088585513
##  [7,]  0.0054434431 -0.0038182673  0.0031180789 -0.0138412293  0.0056422426
##  [8,]  0.0049529477  0.0048398081  0.0112551181  0.0108639823  0.0061588453
##  [9,]  0.0070002297 -0.0023487351  0.0091369153  0.0016206451  0.0026977396
## [10,] -0.0083788455  0.0045658150  0.0100107699  0.0057058802  0.0074822064
## [11,]  0.0256608120 -0.0138128212  0.0084578950 -0.0018403275  0.0009136342
## [12,] -0.0045015574  0.0225132401  0.0065535636  0.0107678216  0.0021290962
## [13,] -0.1973703212 -0.0084972636  0.0032961168 -0.0011685685  0.0045865335
## [14,] -0.0036784659  0.0229421640  0.0246423003  0.0148523490  0.0278851731
## [15,]  0.0131160500 -0.0377273136  0.0073635222 -0.0103747110 -0.0240587188
## [16,]  0.0023938143 -0.0392837206 -0.0610547875 -0.0493787399 -0.0583664498
## [17,]  0.0011181282  0.0042853457 -0.0271053744 -0.0172296043 -0.0110431312
## [18,]  0.0028572920 -0.1881766846  0.0178245506 -0.0066161324 -0.0264523774
## [19,]  0.0101625571 -0.0129195896 -0.1057697631  0.0061481908 -0.0094468771
## [20,]  0.0024071431 -0.0074190766  0.0350454501 -0.1990578033  0.0268459632
## [21,]  0.0031804225 -0.0203083083 -0.0224161773  0.0144628968 -0.1161079657
## [22,]  0.0019043145 -0.0123786542 -0.0115118234 -0.0060163439  0.0239487975
## [23,]  0.0022494496 -0.0099640980 -0.0128817827  0.0066027496 -0.0298600865
## [24,]  0.0023584972 -0.0139905382 -0.0167454048 -0.0032187651 -0.0198359819
## [25,]  0.0082144734 -0.0136767459 -0.0103495015 -0.0069998486 -0.0146234040
## [26,] -0.0116172940 -0.0040655100 -0.0023071203  0.0028607637 -0.0023139799
## [27,]  0.0281644702 -0.0145387940 -0.0025461081  0.0008566042 -0.0020187846
## [28,] -0.0588823014  0.0391825973 -0.0149106121  0.0025096656 -0.0031361669
## [29,] -0.1644537928 -0.0243600305 -0.0006269794 -0.0017683566 -0.0030200354
## [30,]  0.0360479393 -0.1611309530  0.0723154405  0.0060364863  0.0328497116
## [31,] -0.0037277015 -0.0333539952  0.0574855530  0.0397257769 -0.0002090901
## [32,] -0.0058301582  0.0175480637  0.0387726829 -0.1000288108  0.0371083318
## [33,] -0.0064711896  0.0278774517  0.0039996504  0.0602512400 -0.0541461749
## [34,] -0.0087247491  0.0255081700 -0.0088778925  0.0582121923 -0.0091269477
## [35,] -0.0060194242  0.0219885936 -0.0044329648 -0.0045909862  0.0164780310
## [36,] -0.0062771984  0.0137414783 -0.0069243786 -0.0043478259  0.0110544997
## [37,]  0.0008766779  0.0076610747  0.0013412928  0.0050785208  0.0055289013
## [38,]  0.0093039058  0.0191867878 -0.0023982167  0.0061067638  0.0139532744
## [39,]  0.0043709179  0.0037241314 -0.0104139819 -0.0026199086 -0.0001282833
## [40,]  0.0315782011  0.0351654540  0.0204148280  0.0259879821  0.0303275934
## [41,]  0.7906328829 -0.0090991424 -0.0004663476 -0.0015685422 -0.0036983535
## [42,] -0.0090991424  0.2611470151 -0.0237345391  0.0431536339  0.0376301583
## [43,] -0.0004663476 -0.0237345391  0.1273715731  0.0141829098  0.0690300156
## [44,] -0.0015685422  0.0431536339  0.0141829098  0.2504227096 -0.0298783911
## [45,] -0.0036983535  0.0376301583  0.0690300156 -0.0298783911  0.1381497527
## [46,] -0.0010233711  0.0315436576  0.0453536110  0.0659609656  0.0114088058
## [47,] -0.0009052001  0.0321241759  0.0604323940  0.0051091695  0.0844862629
## [48,] -0.0047279510  0.0286513893  0.0573958000  0.0434169573  0.0497803782
## [49,]  0.0201428527  0.0197012819  0.0398304194  0.0285666312  0.0349405049
## [50,] -0.0135538082  0.0089525724  0.0364335660  0.0270941829  0.0252367566
## [51,]  0.0172726808  0.0171765953 -0.0009702544  0.0034730586 -0.0020339755
## [52,] -0.0658950641 -0.0008437656  0.1029914284  0.0720679525  0.0841609132
## [53,] -0.1289854441  0.0179250613 -0.0290966706 -0.0143060475 -0.0206283323
## [54,] -0.0235750772  0.0907859378 -0.0040740589  0.0148474894  0.0043890694
## [55,] -0.0002980855  0.0058543673  0.0112926244  0.0113763222  0.0142109303
## [56,] -0.0061108714  0.0057622944  0.0026763482  0.0210058737  0.0370839900
## [57,] -0.0013913243 -0.0032547701  0.0277091065  0.0270824739  0.0177076040
## [58,] -0.0041414443 -0.0036338212  0.0224847733 -0.0633622109  0.0565608937
## [59,] -0.0008269600  0.0055246973  0.0319829395  0.0824271089 -0.0157446245
## [60,] -0.0151791441  0.0069057207  0.0275921202  0.0199045356  0.0203229074
## [61,]  0.0264467467  0.0151169449  0.0290762396  0.0267648064  0.0239942994
## [62,] -0.0111401736  0.0131687156  0.0363387032  0.0277186199  0.0277813897
## [63,] -0.0189214878  0.0197069877  0.0273676951  0.0212877543  0.0208860300
## [64,]  0.0438253834 -0.0088995917  0.0656016927  0.0461701105  0.0449483808
## [65,] -0.1553056334  0.0381931443  0.0311714639  0.0282118198  0.0332246443
## [66,]  0.0008584699  0.0015101964 -0.0022162731 -0.0004286797 -0.0014353410
##              [,46]         [,47]         [,48]         [,49]        [,50]
##  [1,] -0.005395781 -0.0020300695 -0.0092344195 -0.0035202334 -0.006783795
##  [2,]  0.005003123  0.0017629535  0.0052732048  0.0010156129 -0.001460957
##  [3,]  0.002905069  0.0060188433  0.0090208998  0.0032129542  0.004794528
##  [4,]  0.007664331 -0.0067142608  0.0032819502 -0.0004242595  0.002922520
##  [5,] -0.004236789  0.0142903740  0.0098011625  0.0031102082  0.002833388
##  [6,] -0.016902176  0.0080231452  0.0041298556  0.0032938360  0.002470836
##  [7,]  0.013764859  0.0143629746  0.0225009050 -0.0016336666  0.001563866
##  [8,]  0.011164740  0.0016509927 -0.0201310705 -0.0186461991 -0.001696989
##  [9,]  0.003030303  0.0049870597 -0.0100192511  0.0272996454  0.004229534
## [10,]  0.007340643  0.0029090000  0.0162104859 -0.0149281682  0.003224213
## [11,]  0.002016070  0.0022566350  0.0042759070  0.0104506381  0.006626116
## [12,]  0.008019648  0.0021293934  0.0089601509  0.0068501448 -0.003787253
## [13,]  0.001777476  0.0021385677 -0.0006988548  0.0040177734 -0.014230742
## [14,]  0.014615875  0.0180893930  0.0136571212  0.0098287613 -0.013957204
## [15,] -0.002747691 -0.0072691130  0.0073141001  0.0078064506  0.026834764
## [16,] -0.046530323 -0.0535883254 -0.0516204716 -0.0355965208 -0.028258693
## [17,] -0.017675518 -0.0200814415 -0.0258775896 -0.0211591601 -0.034025921
## [18,] -0.010844243 -0.0175144129 -0.0100333134 -0.0076982738  0.013270472
## [19,] -0.011947314  0.0055311480 -0.0024332523 -0.0004583106  0.018207397
## [20,]  0.004605648 -0.0219545646 -0.0037737394 -0.0039689096  0.013855522
## [21,] -0.024673353  0.0299054286 -0.0089346925 -0.0014179692  0.013658969
## [22,] -0.207527680  0.0418226768 -0.0080802439 -0.0128995410  0.016698204
## [23,]  0.055682050 -0.2160156494  0.0020139298  0.0387696122  0.004506235
## [24,]  0.016534080  0.0041046787 -0.1933558282  0.0272866921  0.017093702
## [25,] -0.012579875 -0.0096217584  0.0349063248 -0.1724471170  0.007364330
## [26,]  0.002290362  0.0002723998  0.0245697281 -0.0331378364 -0.202886039
## [27,] -0.002151566 -0.0010527307 -0.0160021861  0.0278011417 -0.078075911
## [28,]  0.002570377  0.0041816978 -0.0010638109  0.0208304321  0.004167476
## [29,] -0.004669246 -0.0038088073 -0.0061548396  0.0061745147  0.009241787
## [30,]  0.009475832  0.0228730092  0.0116465968  0.0054855285 -0.002614397
## [31,]  0.013474442 -0.0027339978  0.0054142827  0.0032139623 -0.005710714
## [32,]  0.014617084  0.0059866825  0.0087015150  0.0040144802 -0.001565302
## [33,]  0.015173116 -0.0306660769  0.0028416693 -0.0013016668 -0.010674130
## [34,] -0.004448610  0.0433685783 -0.0119017732 -0.0042483479 -0.014042466
## [35,]  0.022129438  0.0337413271 -0.0037170661 -0.0150228002 -0.018841303
## [36,]  0.020972830 -0.0064659771  0.0853664887  0.0251294321 -0.012487906
## [37,]  0.001659987  0.0078639925  0.0294693894  0.0582375776  0.011059056
## [38,]  0.003005047  0.0010369956 -0.0043965955 -0.0410975021  0.036505586
## [39,] -0.004218342 -0.0042234600 -0.0057877928 -0.0148708104  0.059136159
## [40,]  0.020372994  0.0265306873  0.0207847767  0.0221758788  0.035949314
## [41,] -0.001023371 -0.0009052001 -0.0047279510  0.0201428527 -0.013553808
## [42,]  0.031543658  0.0321241759  0.0286513893  0.0197012819  0.008952572
## [43,]  0.045353611  0.0604323940  0.0573958000  0.0398304194  0.036433566
## [44,]  0.065960966  0.0051091695  0.0434169573  0.0285666312  0.027094183
## [45,]  0.011408806  0.0844862629  0.0497803782  0.0349405049  0.025236757
## [46,]  0.358258493 -0.1496445960  0.0528285001  0.0225428909  0.028598827
## [47,] -0.149644596  0.3192144638  0.0257923045  0.0430255613  0.026417370
## [48,]  0.052828500  0.0257923045  0.3169419000 -0.1718348055  0.054321955
## [49,]  0.022542891  0.0430255613 -0.1718348055  0.6763714715 -0.057233394
## [50,]  0.028598827  0.0264173695  0.0543219545 -0.0572333936  0.654395308
## [51,]  0.003044136  0.0030402548 -0.0015644290  0.0159402421 -0.100479761
## [52,]  0.069205664  0.0798331810  0.0749352046  0.0595333253  0.012660485
## [53,] -0.015708172 -0.0166866948 -0.0160439864 -0.0046979260  0.016498352
## [54,]  0.012789954  0.0123522242  0.0171596967  0.0146182158  0.025322152
## [55,]  0.015965881  0.0202348331  0.0243543786  0.0198562595  0.030457186
## [56,]  0.011620149  0.0410364198  0.0187667384  0.0153271583  0.020498770
## [57,]  0.005799196  0.0228537375  0.0234978511  0.0205642873  0.032712394
## [58,]  0.137725070 -0.0414742181  0.0274628856  0.0133423606  0.026515569
## [59,] -0.067450580  0.0084573356  0.0309644345  0.0287072057  0.030702540
## [60,]  0.017520064  0.0238863440 -0.1034137680  0.0660061992  0.054924612
## [61,]  0.023610218  0.0315945895  0.0900502351 -0.1386471692 -0.069641352
## [62,]  0.025944166  0.0315356373  0.0261503556  0.0436513486 -0.243861444
## [63,]  0.019998992  0.0247254570  0.0251750874  0.0375424668  0.062863817
## [64,]  0.044093234  0.0513895031  0.0607927629  0.0250500545  0.074574623
## [65,]  0.027359200  0.0271851336  0.0371376859 -0.0280714496  0.017739256
## [66,]  0.001003138 -0.0011074031 -0.0033948821 -0.0001775879 -0.001943416
##               [,51]         [,52]         [,53]         [,54]         [,55]
##  [1,] -2.364585e-04 -0.0027671840  3.821541e-03  0.0042945537  0.0064417892
##  [2,] -3.457361e-03 -0.0029089369 -3.945616e-03 -0.0025510122 -0.0060657289
##  [3,] -2.922482e-03  0.0084403319 -5.204399e-03 -0.0050286015  0.0049224436
##  [4,]  5.355254e-06 -0.0037848615 -3.496736e-04 -0.0081622644 -0.0017888298
##  [5,] -3.653583e-03  0.0055735956 -4.662755e-03 -0.0087720505 -0.0181082718
##  [6,] -2.585977e-03  0.0045710181 -4.173075e-03 -0.0096883385 -0.0075183641
##  [7,] -5.435892e-03 -0.0009245620  4.473532e-04 -0.0088444920 -0.0122142541
##  [8,] -1.437877e-03  0.0084097174 -7.491182e-03 -0.0114954856 -0.0127563604
##  [9,] -2.854375e-03  0.0038813864 -1.297237e-03 -0.0066476782 -0.0083237355
## [10,] -2.317383e-02 -0.0031713167 -9.753156e-03 -0.0115796947 -0.0131201275
## [11,]  1.789501e-03  0.0065478130  1.664096e-02 -0.0060094586 -0.0017450861
## [12,] -1.090782e-03  0.0083581237 -2.700902e-02 -0.0103528820 -0.0263253461
## [13,]  1.484821e-02 -0.0207728677 -4.007133e-02 -0.0075557934 -0.0009902677
## [14,] -1.444535e-02  0.0137335752 -2.039888e-02 -0.0146406521 -0.0102226959
## [15,]  2.056410e-02 -0.0112652160 -2.484728e-03  0.0357976537  0.0324782594
## [16,] -6.047327e-04 -0.0837585447  2.049140e-02 -0.0102090115 -0.0179530211
## [17,] -2.581822e-02 -0.0218082123 -1.043094e-02 -0.0374301127 -0.0457783057
## [18,] -1.079494e-02  0.0484875770 -1.380109e-02 -0.0211102545  0.0253386903
## [19,]  1.714227e-02 -0.0059452985  1.615959e-02  0.0075658279 -0.0088998758
## [20,]  1.173135e-02 -0.0037032315  1.268247e-02  0.0196640620  0.0168603364
## [21,]  1.286864e-02 -0.0054387827  1.647504e-02  0.0129779085  0.0091313902
## [22,]  1.056399e-02 -0.0009103506  1.398667e-02  0.0133187627  0.0112334813
## [23,]  1.133465e-02  0.0010466791  1.228483e-02  0.0066855946  0.0053136782
## [24,]  7.290832e-03 -0.0059426914  1.538149e-02  0.0131984076  0.0091110676
## [25,]  4.123379e-02  0.0071490221  6.692713e-03  0.0152347729  0.0120273031
## [26,] -6.395511e-02 -0.0070594573  1.955031e-02  0.0069622195  0.0067169958
## [27,] -1.954144e-01 -0.0030352531  2.810298e-03 -0.0008744469 -0.0044089100
## [28,]  2.849550e-02 -0.1437869877  2.988149e-02  0.0104048152  0.0118246155
## [29,]  2.871742e-02  0.0183116814 -1.587029e-01 -0.0053935949 -0.0075879118
## [30,] -1.011215e-02  0.0297959804 -3.186867e-03 -0.1639790787 -0.0030354815
## [31,] -1.084683e-02  0.0285635489 -8.286545e-03 -0.0013752373 -0.1724997665
## [32,] -5.421362e-03  0.0179415407 -8.501486e-04 -0.0151558549  0.0391262767
## [33,] -9.145045e-03  0.0135567431  1.811041e-03 -0.0156134073 -0.0071904981
## [34,] -9.945932e-03  0.0052048370  3.659841e-03 -0.0204977801 -0.0177993173
## [35,] -1.139554e-02  0.0066785728  4.194751e-03 -0.0211956602 -0.0202134958
## [36,] -3.046121e-03 -0.0008406137  5.669235e-03 -0.0134369816 -0.0123020375
## [37,] -1.221532e-02 -0.0003688363  7.027523e-03 -0.0042793422 -0.0018783211
## [38,]  3.295220e-02  0.0205736754 -2.721699e-04 -0.0160149979 -0.0128849221
## [39,] -2.356010e-02 -0.0147248087  4.833602e-05 -0.0287176966 -0.0111602699
## [40,]  9.704547e-03 -0.0038231824 -6.233230e-02  0.0420412524  0.0063231547
## [41,]  1.727268e-02 -0.0658950641 -1.289854e-01 -0.0235750772 -0.0002980855
## [42,]  1.717660e-02 -0.0008437656  1.792506e-02  0.0907859378  0.0058543673
## [43,] -9.702544e-04  0.1029914284 -2.909667e-02 -0.0040740589  0.0112926244
## [44,]  3.473059e-03  0.0720679525 -1.430605e-02  0.0148474894  0.0113763222
## [45,] -2.033975e-03  0.0841609132 -2.062833e-02  0.0043890694  0.0142109303
## [46,]  3.044136e-03  0.0692056643 -1.570817e-02  0.0127899536  0.0159658806
## [47,]  3.040255e-03  0.0798331810 -1.668669e-02  0.0123522242  0.0202348331
## [48,] -1.564429e-03  0.0749352046 -1.604399e-02  0.0171596967  0.0243543786
## [49,]  1.594024e-02  0.0595333253 -4.697926e-03  0.0146182158  0.0198562595
## [50,] -1.004798e-01  0.0126604854  1.649835e-02  0.0253221516  0.0304571862
## [51,]  6.167309e-01  0.1315020669 -8.079103e-02  0.0220609689  0.0248279390
## [52,]  1.315021e-01  0.4806127845  2.026605e-01  0.0097030183  0.0188968315
## [53,] -8.079103e-02  0.2026605310  6.443417e-01  0.0115061731  0.0134090071
## [54,]  2.206097e-02  0.0097030183  1.150617e-02  0.0850293918  0.0272958907
## [55,]  2.482794e-02  0.0188968315  1.340901e-02  0.0272958907  0.0902043856
## [56,]  1.667404e-02  0.0182099867  1.159331e-02  0.0240299823  0.0097499188
## [57,]  2.534125e-02  0.0214998205  1.078421e-02  0.0360793898  0.0466459021
## [58,]  1.947496e-02  0.0182524204  1.097019e-02  0.0266888234  0.0303325993
## [59,]  2.429652e-02  0.0302483148  7.856209e-03  0.0366161895  0.0495100704
## [60,]  2.269610e-02  0.0270865861  4.576444e-03  0.0274959916  0.0367272774
## [61,]  7.926892e-03  0.0380416906  1.529663e-02  0.0196686469  0.0280824220
## [62,]  7.085788e-02  0.0524863820 -1.093323e-02  0.0265726851  0.0359360810
## [63,] -2.436638e-01 -0.0268352772  4.729051e-02  0.0282508523  0.0186877405
## [64,] -5.330921e-02 -0.0805112665 -9.540525e-02  0.0296482945  0.0862217453
## [65,] -8.524540e-03 -0.0896372031 -2.768191e-01  0.0216935400  0.0046342298
## [66,]  1.605257e-03 -0.0021589535 -6.288517e-04  0.0004718029 -0.0012138514
##               [,56]         [,57]         [,58]         [,59]         [,60]
##  [1,]  0.0133772647  0.0030035079  0.0086847373  0.0028009521  0.0097629109
##  [2,] -0.0239182572 -0.0115227812 -0.0119222143 -0.0126099530 -0.0143443970
##  [3,]  0.0015898082 -0.0084596966 -0.0061553781 -0.0084590441 -0.0097046036
##  [4,] -0.0052166455 -0.0081458516 -0.0174081674 -0.0025718057 -0.0127554958
##  [5,]  0.0119449599  0.0070702220  0.0017581762 -0.0231336044 -0.0101778200
##  [6,] -0.0201079366 -0.0055876135 -0.0049297443  0.0118014677 -0.0148932648
##  [7,]  0.0019033333 -0.0202867394  0.0185312593 -0.0076327749 -0.0082735509
##  [8,] -0.0183787069 -0.0077161968 -0.0221427199 -0.0057538018  0.0203518922
##  [9,] -0.0119988180 -0.0073193020 -0.0072827997 -0.0089842000  0.0051733678
## [10,] -0.0140774929 -0.0113459168 -0.0147321171 -0.0089623848 -0.0162953866
## [11,] -0.0095887018 -0.0009120609 -0.0029671368 -0.0014373738 -0.0073458634
## [12,] -0.0219634095 -0.0213474098 -0.0264240281 -0.0193702352 -0.0259678477
## [13,] -0.0020101887 -0.0028917952 -0.0002418905 -0.0038121273 -0.0033329734
## [14,] -0.0114688384 -0.0116544096 -0.0136186439 -0.0007215817 -0.0059462236
## [15,]  0.0018710024  0.0440419804  0.0221775652  0.0415663433  0.0155785247
## [16,] -0.0149137656 -0.0168098989 -0.0139093752 -0.0249579251 -0.0223242279
## [17,] -0.0287688242 -0.0495195041 -0.0372089710 -0.0501930591 -0.0379921136
## [18,]  0.0165779206  0.0162950299  0.0099525890  0.0117159623  0.0112608763
## [19,]  0.0220283358  0.0130501084  0.0341609078 -0.0049922873  0.0106829226
## [20,] -0.0315615880  0.0207189220  0.0195169423  0.0013617384  0.0062238459
## [21,]  0.0219066469 -0.0021539325 -0.0065761205  0.0093399196  0.0095025959
## [22,]  0.0088413115  0.0137202703 -0.0432118615  0.0656450057  0.0025855639
## [23,] -0.0028922385  0.0068521783  0.0120773017  0.0397893353  0.0314630933
## [24,]  0.0090327129  0.0111459749  0.0191884403 -0.0006987365  0.0372453554
## [25,]  0.0104888489  0.0145347151  0.0095007730  0.0107786158 -0.0125694862
## [26,]  0.0103639016  0.0054394425  0.0107573567 -0.0014292657  0.0111757557
## [27,]  0.0018062115 -0.0055588092 -0.0035114542 -0.0055743006 -0.0118011128
## [28,]  0.0180972314  0.0111261407  0.0196117530 -0.0003622190  0.0036729331
## [29,] -0.0032963732 -0.0092675959 -0.0077917616 -0.0073857574 -0.0130435515
## [30,]  0.0136204137 -0.0169333533  0.0102692145 -0.0225625949  0.0007983492
## [31,] -0.0040417009 -0.0019874327 -0.0451495758  0.0227605664 -0.0010815190
## [32,] -0.2427388081  0.0204718055 -0.0180596044  0.0027099528  0.0007178985
## [33,] -0.0534336104 -0.1302139802 -0.0315718023  0.0346814437 -0.0041288369
## [34,] -0.0369788535  0.0344121640 -0.1937802574  0.0158351936 -0.0092962666
## [35,]  0.0096030609 -0.0426980726  0.0435557977 -0.2003819682  0.0162174918
## [36,]  0.0092025187 -0.0194340680  0.0110094230  0.0004513667 -0.2433983317
## [37,]  0.0015176594 -0.0043876149 -0.0012328970 -0.0022332963 -0.0324212208
## [38,]  0.0034582847 -0.0191290900 -0.0072104379 -0.0219185454  0.0426194395
## [39,] -0.0034371517 -0.0165224391 -0.0091118197 -0.0171922876 -0.0094125335
## [40,]  0.0230885737  0.0050780871  0.0143494669  0.0055056135  0.0112065215
## [41,] -0.0061108714 -0.0013913243 -0.0041414443 -0.0008269600 -0.0151791441
## [42,]  0.0057622944 -0.0032547701 -0.0036338212  0.0055246973  0.0069057207
## [43,]  0.0026763482  0.0277091065  0.0224847733  0.0319829395  0.0275921202
## [44,]  0.0210058737  0.0270824739 -0.0633622109  0.0824271089  0.0199045356
## [45,]  0.0370839900  0.0177076040  0.0565608937 -0.0157446245  0.0203229074
## [46,]  0.0116201491  0.0057991963  0.1377250696 -0.0674505803  0.0175200640
## [47,]  0.0410364198  0.0228537375 -0.0414742181  0.0084573356  0.0238863440
## [48,]  0.0187667384  0.0234978511  0.0274628856  0.0309644345 -0.1034137680
## [49,]  0.0153271583  0.0205642873  0.0133423606  0.0287072057  0.0660061992
## [50,]  0.0204987704  0.0327123942  0.0265155695  0.0307025404  0.0549246118
## [51,]  0.0166740363  0.0253412546  0.0194749602  0.0242965208  0.0226960967
## [52,]  0.0182099867  0.0214998205  0.0182524204  0.0302483148  0.0270865861
## [53,]  0.0115933050  0.0107842055  0.0109701863  0.0078562095  0.0045764445
## [54,]  0.0240299823  0.0360793898  0.0266888234  0.0366161895  0.0274959916
## [55,]  0.0097499188  0.0466459021  0.0303325993  0.0495100704  0.0367272774
## [56,]  0.1766275305  0.0016888738  0.1042098777 -0.0311666667  0.0288868134
## [57,]  0.0016888738  0.0930757262 -0.0089453172  0.0746626199  0.0368648161
## [58,]  0.1042098777 -0.0089453172  0.2283859179 -0.0904742279  0.0315720899
## [59,] -0.0311666667  0.0746626199 -0.0904742279  0.2204526680  0.0210437160
## [60,]  0.0288868134  0.0368648161  0.0315720899  0.0210437160  0.2161986345
## [61,]  0.0210037763  0.0277967468  0.0225735623  0.0378442013 -0.0849599736
## [62,]  0.0257941566  0.0369387847  0.0289825723  0.0412909794  0.0287777190
## [63,]  0.0128227547  0.0213395358  0.0159285577  0.0252632679  0.0030211498
## [64,]  0.0595274990  0.0845605109  0.0675441666  0.0899132765  0.0855541182
## [65,]  0.0083738747  0.0054862653  0.0068498269  0.0066021606  0.0370346926
## [66,] -0.0007118678 -0.0006393511 -0.0007113496 -0.0014419246 -0.0014379877
##               [,61]         [,62]         [,63]         [,64]         [,65]
##  [1,]  0.0104048075  0.0057277255 -0.0034190627  0.0209601764  0.0003528548
##  [2,] -0.0115929275 -0.0093742710 -0.0089434166 -0.0085640365 -0.0113283603
##  [3,] -0.0094981683 -0.0065709680  0.0062413979 -0.0323566992  0.0067236209
##  [4,] -0.0118407336 -0.0093699579 -0.0016844081 -0.0239839885 -0.0026815358
##  [5,] -0.0101530516 -0.0078859861  0.0019287352 -0.0266084112  0.0012885121
##  [6,] -0.0110789158 -0.0078355277  0.0000880042 -0.0233362959  0.0024200786
##  [7,] -0.0128012130 -0.0096364342 -0.0003335098 -0.0283046940 -0.0031414824
##  [8,]  0.0151154185 -0.0086586454  0.0021410025 -0.0263816842  0.0025386953
##  [9,] -0.0149850240 -0.0013658672 -0.0049521897 -0.0254259202 -0.0009411538
## [10,] -0.0028197219 -0.0074101978  0.0182957630 -0.0111545846  0.0163542002
## [11,] -0.0016953214  0.0052562254  0.0159555744 -0.0209816906 -0.0122245356
## [12,] -0.0145975130 -0.0163019787 -0.0112848783 -0.0302882739  0.0138752999
## [13,]  0.0064137620 -0.0017361889 -0.0101745216  0.0099379242  0.0517181975
## [14,]  0.0043906048  0.0072477418  0.0052925822  0.0045919993  0.0104689558
## [15,]  0.0003722837  0.0176506554  0.0190139896  0.0347583997 -0.0104446211
## [16,] -0.0269417009 -0.0311561785 -0.0238484589 -0.0514275297 -0.0325452235
## [17,] -0.0275582711 -0.0376627339 -0.0224723303 -0.0852799873 -0.0057662276
## [18,]  0.0015565738 -0.0012103870 -0.0086415036  0.0248867928 -0.0169423414
## [19,]  0.0006178859  0.0017049453 -0.0023993601  0.0185190816 -0.0104708540
## [20,] -0.0005272086 -0.0010980369 -0.0000041520  0.0022506025 -0.0038647662
## [21,] -0.0021165105 -0.0033454646 -0.0021546259  0.0007934607 -0.0076409833
## [22,]  0.0014785290 -0.0025196270 -0.0010268543  0.0036393855 -0.0031114839
## [23,] -0.0198273935 -0.0034375237 -0.0043659735 -0.0058648356 -0.0050388216
## [24,] -0.0089135226 -0.0057299641 -0.0016010097  0.0006101185 -0.0043061557
## [25,]  0.0305695904  0.0120083138 -0.0139470508 -0.0033851188 -0.0116242894
## [26,] -0.0130703166  0.0318301289  0.0582830426  0.0384262139  0.0251594869
## [27,]  0.0293780376  0.0575276360  0.0515255763 -0.0030165825 -0.0111199347
## [28,]  0.0284116767  0.0157411733  0.0037408808  0.0263094176 -0.0001743791
## [29,]  0.0161001550 -0.0217201417 -0.0203009885 -0.0169106353  0.0243429041
## [30,]  0.0063774196  0.0001260029 -0.0162730283  0.0355395545 -0.0082050518
## [31,]  0.0074815475  0.0012658243 -0.0067502962  0.0114441878  0.0023271649
## [32,]  0.0057906434  0.0014278514 -0.0006726951  0.0035852475  0.0084658492
## [33,]  0.0074810581 -0.0037851759 -0.0068478491 -0.0067473775  0.0069955757
## [34,] -0.0041560291 -0.0105085502 -0.0129388328 -0.0181531198  0.0074794065
## [35,]  0.0176574177 -0.0102769940 -0.0094127367 -0.0229138158  0.0018988666
## [36,] -0.0089762188 -0.0058876473 -0.0111654676 -0.0121429037  0.0030129565
## [37,] -0.1877971855 -0.0021033168  0.0085965603 -0.0010731077 -0.0125180928
## [38,] -0.1145288190 -0.1705208566  0.0154143475  0.0059237874  0.0168483703
## [39,] -0.0063287301  0.0048782454 -0.1457908340 -0.0079128852 -0.0701170375
## [40,]  0.0370688781  0.0160272515  0.0455735248 -0.1904742790  0.0496067440
## [41,]  0.0264467467 -0.0111401736 -0.0189214878  0.0438253834 -0.1553056334
## [42,]  0.0151169449  0.0131687156  0.0197069877 -0.0088995917  0.0381931443
## [43,]  0.0290762396  0.0363387032  0.0273676951  0.0656016927  0.0311714639
## [44,]  0.0267648064  0.0277186199  0.0212877543  0.0461701105  0.0282118198
## [45,]  0.0239942994  0.0277813897  0.0208860300  0.0449483808  0.0332246443
## [46,]  0.0236102184  0.0259441664  0.0199989920  0.0440932341  0.0273591995
## [47,]  0.0315945895  0.0315356373  0.0247254570  0.0513895031  0.0271851336
## [48,]  0.0900502351  0.0261503556  0.0251750874  0.0607927629  0.0371376859
## [49,] -0.1386471692  0.0436513486  0.0375424668  0.0250500545 -0.0280714496
## [50,] -0.0696413515 -0.2438614437  0.0628638169  0.0745746233  0.0177392563
## [51,]  0.0079268925  0.0708578811 -0.2436637980 -0.0533092148 -0.0085245404
## [52,]  0.0380416906  0.0524863820 -0.0268352772 -0.0805112665 -0.0896372031
## [53,]  0.0152966323 -0.0109332287  0.0472905063 -0.0954052477 -0.2768190588
## [54,]  0.0196686469  0.0265726851  0.0282508523  0.0296482945  0.0216935400
## [55,]  0.0280824220  0.0359360810  0.0186877405  0.0862217453  0.0046342298
## [56,]  0.0210037763  0.0257941566  0.0128227547  0.0595274990  0.0083738747
## [57,]  0.0277967468  0.0369387847  0.0213395358  0.0845605109  0.0054862653
## [58,]  0.0225735623  0.0289825723  0.0159285577  0.0675441666  0.0068498269
## [59,]  0.0378442013  0.0412909794  0.0252632679  0.0899132765  0.0066021606
## [60,] -0.0849599736  0.0287777190  0.0030211498  0.0855541182  0.0370346926
## [61,]  0.4418794068  0.0444534000  0.0583695799  0.0357131473 -0.0649138742
## [62,]  0.0444534000  0.2133387851 -0.0287962759  0.0718289366  0.0397616242
## [63,]  0.0583695799 -0.0287962759  0.1794701862  0.0511023818  0.0086363648
## [64,]  0.0357131473  0.0718289366  0.0511023818  0.3559968863  0.0913891662
## [65,] -0.0649138742  0.0397616242  0.0086363648  0.0913891662  0.2765623985
## [66,]  0.0001908403  0.0000509350 -0.0017348315 -0.0020392014 -0.0008992686
##               [,66]
##  [1,]  1.158708e-03
##  [2,] -1.972937e-04
##  [3,] -1.527658e-03
##  [4,] -4.302898e-04
##  [5,] -1.951212e-03
##  [6,] -1.737774e-03
##  [7,] -1.939218e-03
##  [8,] -5.929875e-04
##  [9,] -1.645049e-03
## [10,] -3.281460e-04
## [11,] -2.822462e-03
## [12,] -1.529064e-03
## [13,]  1.338725e-03
## [14,]  9.945125e-04
## [15,]  6.510796e-04
## [16,]  1.170030e-03
## [17,]  7.830403e-04
## [18,] -3.350708e-03
## [19,]  4.893545e-04
## [20,] -1.856978e-04
## [21,] -2.066725e-03
## [22,] -2.289379e-03
## [23,] -3.596608e-03
## [24,]  1.722355e-03
## [25,]  2.473749e-03
## [26,] -6.296412e-03
## [27,]  2.225938e-03
## [28,] -6.461750e-04
## [29,] -2.058868e-03
## [30,] -4.137158e-03
## [31,]  1.297036e-03
## [32,] -2.637089e-03
## [33,] -8.338188e-05
## [34,]  2.255103e-04
## [35,]  7.055726e-04
## [36,]  1.426239e-03
## [37,] -5.401815e-04
## [38,] -1.610907e-03
## [39,]  1.325791e-03
## [40,] -3.238743e-03
## [41,]  8.584699e-04
## [42,]  1.510196e-03
## [43,] -2.216273e-03
## [44,] -4.286797e-04
## [45,] -1.435341e-03
## [46,]  1.003138e-03
## [47,] -1.107403e-03
## [48,] -3.394882e-03
## [49,] -1.775879e-04
## [50,] -1.943416e-03
## [51,]  1.605257e-03
## [52,] -2.158954e-03
## [53,] -6.288517e-04
## [54,]  4.718029e-04
## [55,] -1.213851e-03
## [56,] -7.118678e-04
## [57,] -6.393511e-04
## [58,] -7.113496e-04
## [59,] -1.441925e-03
## [60,] -1.437988e-03
## [61,]  1.908403e-04
## [62,]  5.093500e-05
## [63,] -1.734832e-03
## [64,] -2.039201e-03
## [65,] -8.992686e-04
## [66,]  5.342271e-03
## 
## $log_evidence
## [1] -160.9409
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 148

3g)

Use the viz_post_coefs() function to visualize the posterior coefficient summaries for model 3 and model 6, based on the strong prior specification.

SOLUTION

### add more code chunks if you like
viz_post_coefs(laplace_03_strong$mode[-length(laplace_03_strong$mode)],
               sqrt(diag(laplace_03_strong$var_matrix))[-length(laplace_03_strong$mode)],
               info_03_strong$design_matrix %>% colnames())

viz_post_coefs(laplace_06_strong$mode[-length(laplace_06_strong$mode)],
               sqrt(diag(laplace_06_strong$var_matrix))[-length(laplace_06_strong$mode)],
               info_06_strong$design_matrix %>% colnames())

3h)

You will fit one more set of Bayesian models with a very strong prior on the regression coefficients. The prior standard deviation will be equal to 1/50.

Complete the first code chunk below, which defines the list of required information for both the model 3 and model 6 formulations using the very strong prior on the regression coefficients. All other information, data and the \(\sigma\) prior, are the same as before.

Run the Laplace Approximation using the strong prior for both the model 3 and model 6 formulations. Assign the results to laplace_03_very_strong and laplace_06_very_strong.

Confirm that the optimizations converged for both laplace approximation results.

SOLUTION

info_03_very_strong <- list(
  yobs = df$y,
  design_matrix = info_03_weak$design_matrix,
  mu_beta = 0,
  tau_beta = 1/50,
  sigma_rate = 1
)

info_06_very_strong <- list(
  yobs = df$y,
  design_matrix = info_06_weak$design_matrix,
  mu_beta = 0,
  tau_beta = 1/50,
  sigma_rate = 1
)

Execute the Laplace Approximation.

### add more code chunks if you like
laplace_03_very_strong <- my_laplace(start_guess_03,
                                     lm_logpost,
                                     info_03_very_strong)
laplace_03_very_strong

## $mode
##  [1]  0.003592761  0.006873296 -0.003563924 -0.005548833 -0.034591615
##  [6] -0.005596963  0.006696281 -0.010922248 -0.036247920 -0.082665953
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  3.845484e-04  2.590456e-07 -1.379063e-05 -2.342351e-07 -1.718064e-05
##  [2,]  2.590456e-07  3.840058e-04  3.516814e-06 -1.839586e-06 -3.016308e-06
##  [3,] -1.379063e-05  3.516814e-06  3.644930e-04 -1.862478e-06 -1.159401e-05
##  [4,] -2.342351e-07 -1.839586e-06 -1.862478e-06  3.837901e-04  2.426882e-06
##  [5,] -1.718064e-05 -3.016308e-06 -1.159401e-05  2.426882e-06  3.747607e-04
##  [6,] -9.397074e-07 -3.102062e-06 -3.791051e-06  2.222885e-06 -5.962896e-06
##  [7,]  1.343448e-06 -1.574857e-05  6.109602e-06 -9.945583e-06  2.864426e-06
##  [8,] -3.066295e-06 -6.377984e-06 -2.351645e-06 -1.551120e-05  6.543729e-07
##  [9,] -1.471935e-05  2.996232e-06 -3.702900e-05 -2.145655e-06 -2.186027e-05
## [10,] -6.498839e-05 -6.513481e-05 -1.287717e-05  5.111098e-05  3.033539e-04
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,] -9.397074e-07  1.343448e-06 -3.066295e-06 -1.471935e-05 -6.498839e-05
##  [2,] -3.102062e-06 -1.574857e-05 -6.377984e-06  2.996232e-06 -6.513481e-05
##  [3,] -3.791051e-06  6.109602e-06 -2.351645e-06 -3.702900e-05 -1.287717e-05
##  [4,]  2.222885e-06 -9.945583e-06 -1.551120e-05 -2.145655e-06  5.111098e-05
##  [5,] -5.962896e-06  2.864426e-06  6.543729e-07 -2.186027e-05  3.033539e-04
##  [6,]  3.826461e-04 -6.989653e-06  9.383101e-06 -1.666292e-05  3.746314e-05
##  [7,] -6.989653e-06  3.529481e-04 -2.161346e-05  8.768168e-06 -5.358871e-05
##  [8,]  9.383101e-06 -2.161346e-05  3.545232e-04  2.842741e-06  1.034923e-04
##  [9,] -1.666292e-05  8.768168e-06  2.842741e-06  3.207896e-04  2.649717e-04
## [10,]  3.746314e-05 -5.358871e-05  1.034923e-04  2.649717e-04  5.546760e-03
## 
## $log_evidence
## [1] -140.2491
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 84

laplace_06_very_strong <- my_laplace(start_guess_06,
                                     lm_logpost,
                                     info_06_very_strong)
laplace_06_very_strong

## $mode
##  [1]  9.595320e-03  2.025166e-04  4.505341e-04  1.246627e-03  2.045638e-03
##  [6]  1.145230e-03  2.340198e-04  1.992312e-03  1.798180e-03  9.429094e-04
## [11]  1.639782e-03 -8.279119e-05  2.350166e-04 -7.058068e-03 -1.630164e-02
## [16] -1.277296e-02 -5.098678e-02 -8.853261e-04  9.585182e-04 -1.877547e-04
## [21] -4.061852e-04  2.894994e-04 -2.911555e-04 -1.957401e-03 -1.220925e-03
## [26] -1.859333e-04  5.269584e-05 -1.752511e-03  3.444512e-04 -9.019286e-05
## [31] -4.984301e-04 -1.634918e-04 -3.140493e-03 -2.410296e-03 -1.818257e-03
## [36] -7.205437e-04  3.737394e-05 -5.742500e-04 -2.464914e-04 -2.426018e-03
## [41]  3.902126e-04 -3.532866e-03  3.818524e-03 -3.519940e-04  8.338765e-04
## [46]  1.454893e-03 -1.922879e-03 -2.627591e-03 -1.208161e-03 -1.183203e-03
## [51] -7.897388e-04 -4.436582e-03  7.269596e-04  5.313389e-03  6.979266e-04
## [56] -9.424258e-04 -1.477504e-02 -1.059741e-02 -5.655724e-03 -3.734206e-03
## [61] -1.390676e-03 -2.459181e-03 -2.553076e-03 -6.011259e-03  1.950180e-04
## [66] -1.901450e-01
## 
## $var_matrix
##                [,1]          [,2]          [,3]          [,4]          [,5]
##  [1,]  3.837176e-04 -1.518928e-06 -1.564506e-06 -1.076522e-06 -1.257067e-06
##  [2,] -1.518928e-06  3.991733e-04 -4.444459e-07 -1.235300e-08  5.998136e-08
##  [3,] -1.564506e-06 -4.444459e-07  3.989233e-04 -3.275556e-07  2.570194e-08
##  [4,] -1.076522e-06 -1.235300e-08 -3.275556e-07  3.993239e-04 -3.817015e-07
##  [5,] -1.257067e-06  5.998136e-08  2.570194e-08 -3.817015e-07  3.989651e-04
##  [6,] -1.109564e-06  4.151158e-08  4.573016e-08  4.067442e-08 -3.591925e-07
##  [7,] -1.249922e-06  2.338736e-08  2.507771e-08  2.673983e-08  3.746780e-09
##  [8,] -1.833931e-06  3.847787e-08  4.992998e-08  7.859778e-08  1.358024e-07
##  [9,] -1.249185e-06  2.653786e-08  3.665102e-08  6.375024e-08  1.106793e-07
## [10,] -1.207047e-06  3.376781e-08  4.032391e-08  5.274356e-08  8.853751e-08
## [11,] -2.984415e-07  1.809009e-07  4.782518e-08  6.883375e-08  1.173174e-07
## [12,] -1.676348e-06 -3.308032e-07  1.673559e-08  2.449224e-08  3.773776e-08
## [13,]  3.837901e-07  2.072744e-07  5.203185e-09  2.883411e-09  5.111411e-09
## [14,] -8.551570e-07 -6.831185e-08  2.558705e-07 -3.131077e-07 -8.770093e-08
## [15,] -1.289480e-05 -7.653976e-07 -9.777198e-07 -9.141577e-07 -1.632436e-06
## [16,] -5.381842e-06  3.676791e-07 -2.667817e-07 -4.498030e-07  3.738204e-07
## [17,] -1.674485e-05 -1.778064e-06 -1.544158e-06 -1.242333e-06 -2.595546e-06
## [18,] -5.749400e-07 -6.385854e-08  2.976913e-08 -4.156176e-08 -1.102950e-07
## [19,]  6.688739e-07  1.100974e-07  2.470222e-07  3.683119e-08  3.928000e-08
## [20,] -1.666892e-07  8.784905e-09  4.948827e-10 -1.319863e-07 -7.649760e-08
## [21,] -1.575422e-07 -4.656215e-08 -4.758724e-08 -1.064912e-07  6.655866e-08
## [22,]  2.134833e-07 -2.217272e-08 -1.356755e-08 -7.477695e-09  1.454313e-07
## [23,] -1.467417e-07  6.150377e-09  7.643870e-09  5.032730e-09  1.329346e-08
## [24,]  1.795823e-07 -8.321694e-09 -1.706529e-08 -4.728880e-08 -8.927159e-08
## [25,]  2.112884e-07 -6.419615e-09 -1.238029e-08 -3.167199e-08 -5.811371e-08
## [26,]  2.304404e-07  3.658512e-08  3.793090e-08  3.331888e-08  5.259152e-08
## [27,]  1.415656e-07  7.250357e-08  4.695195e-08  4.325408e-08  7.282597e-08
## [28,] -8.426327e-07 -6.208568e-08 -1.043983e-09 -2.613577e-08 -5.767323e-08
## [29,]  6.627573e-08  4.841503e-08  1.017957e-08  1.242478e-08  2.508919e-08
## [30,] -2.061934e-08 -6.363355e-07 -1.688978e-07  7.497963e-08  2.556779e-07
## [31,] -6.933501e-07 -2.509352e-07 -9.713017e-07 -2.224666e-07  1.283729e-07
## [32,] -6.555488e-07  1.370992e-08 -2.455097e-07 -4.315758e-07 -3.437790e-07
## [33,] -1.679837e-06  1.197064e-07  7.258005e-08 -4.056090e-07 -1.264772e-06
## [34,] -1.648317e-06  7.301378e-08  5.208042e-08 -1.050214e-08 -4.871504e-07
## [35,] -1.488793e-06  2.991577e-08  2.125771e-08 -1.252219e-08 -5.861372e-08
## [36,] -1.485382e-06  2.932432e-08  2.714374e-08  1.059529e-08  1.462491e-08
## [37,] -6.110354e-07  1.763931e-08  1.867346e-08  1.694071e-08  2.823296e-08
## [38,] -4.960866e-07  7.916099e-08  8.089889e-08  6.496812e-08  1.036155e-07
## [39,] -6.117359e-08  1.846734e-07  7.660015e-08  6.633759e-08  1.045342e-07
## [40,] -1.422390e-06 -2.114870e-07  4.939768e-08  5.758143e-09  4.609202e-09
## [41,]  2.121388e-07  1.470179e-07 -9.387142e-09 -3.158845e-09 -8.085157e-09
## [42,] -3.385292e-06  2.224860e-07 -3.589126e-07 -3.513635e-07 -7.246068e-07
## [43,]  1.595862e-06  2.573059e-07  6.559765e-08  8.918269e-08  1.811941e-07
## [44,] -6.455058e-08 -3.008161e-09 -5.889163e-08 -1.107162e-07 -6.295850e-09
## [45,] -3.812199e-08 -1.882367e-07 -1.553674e-07 -9.357741e-08  4.525079e-07
## [46,]  5.648680e-07 -8.474648e-08 -5.261757e-08 -3.402557e-08  5.002210e-07
## [47,] -5.661182e-07  1.430628e-08  1.169984e-08 -1.762835e-08 -1.592823e-08
## [48,] -4.664659e-07  1.326195e-08  3.605053e-09 -3.950586e-08 -8.093377e-08
## [49,] -9.888857e-08  1.076593e-08  6.293541e-09 -1.424979e-08 -3.045218e-08
## [50,] -7.405878e-08  1.558495e-07  1.591564e-07  1.286055e-07  2.046428e-07
## [51,]  2.067333e-07  2.227522e-07  1.948957e-07  1.625305e-07  2.737586e-07
## [52,] -2.037791e-06 -7.381924e-08 -9.530580e-09 -7.110718e-08 -1.640797e-07
## [53,]  3.406248e-07  8.501150e-08  4.121024e-08  4.217995e-08  8.734880e-08
## [54,]  6.652979e-06 -2.006045e-06  3.072726e-07  8.613631e-07  2.060142e-06
## [55,]  5.872295e-07 -6.770527e-07 -2.748992e-06 -3.448906e-07  8.681840e-07
## [56,] -1.225304e-06  7.373955e-08 -6.446816e-07 -9.707759e-07 -8.946744e-07
## [57,] -5.017787e-06  4.354222e-07  2.709191e-07 -1.173723e-06 -3.797297e-06
## [58,] -4.220555e-06  2.367368e-07  1.470570e-07 -8.086236e-08 -1.500183e-06
## [59,] -2.988305e-06  7.223578e-08  4.567180e-08 -5.569282e-08 -1.748094e-07
## [60,] -2.408010e-06  5.281503e-08  3.811824e-08 -3.150115e-08 -6.805816e-08
## [61,] -9.085531e-07  3.640897e-08  3.188060e-08  4.302856e-09  1.141384e-09
## [62,] -4.706613e-07  3.231449e-07  3.295320e-07  2.645382e-07  4.228842e-07
## [63,] -3.944244e-07  5.704071e-07  2.721275e-07  2.190975e-07  3.272147e-07
## [64,] -1.870679e-06 -4.237973e-07  2.945506e-07  1.477507e-07  2.671009e-07
## [65,] -2.623744e-07  3.281596e-07 -1.038818e-07 -8.064689e-08 -1.633992e-07
## [66,] -1.260489e-04 -3.626851e-06 -6.310892e-06 -1.492951e-05 -2.623075e-05
##                [,6]          [,7]          [,8]          [,9]         [,10]
##  [1,] -1.109564e-06 -1.249922e-06 -1.833931e-06 -1.249185e-06 -1.207047e-06
##  [2,]  4.151158e-08  2.338736e-08  3.847787e-08  2.653786e-08  3.376781e-08
##  [3,]  4.573016e-08  2.507771e-08  4.992998e-08  3.665102e-08  4.032391e-08
##  [4,]  4.067442e-08  2.673983e-08  7.859778e-08  6.375024e-08  5.274356e-08
##  [5,] -3.591925e-07  3.746780e-09  1.358024e-07  1.106793e-07  8.853751e-08
##  [6,]  3.992703e-04 -3.651945e-07  2.220361e-08  6.797898e-08  5.640197e-08
##  [7,] -3.651945e-07  3.993192e-04 -3.689930e-07 -1.912230e-09  2.951570e-08
##  [8,]  2.220361e-08 -3.689930e-07  3.986087e-04 -5.014213e-07  3.591678e-08
##  [9,]  6.797898e-08 -1.912230e-09 -5.014213e-07  3.992335e-04 -2.950465e-07
## [10,]  5.640197e-08  2.951570e-08  3.591678e-08 -2.950465e-07  3.992294e-04
## [11,]  7.150613e-08  3.074398e-08  9.557878e-08  5.271740e-08 -3.728258e-07
## [12,]  2.739159e-08  2.264042e-08  2.861010e-08  9.967844e-09 -1.258520e-07
## [13,]  1.895142e-09 -1.451909e-09  5.527929e-09  1.086818e-08  4.179298e-08
## [14,]  9.149868e-08 -2.122490e-07  3.379792e-07  2.604244e-07 -1.477561e-07
## [15,] -1.413283e-06 -1.072105e-06 -1.546715e-06 -8.664735e-07 -8.539284e-07
## [16,]  2.948725e-07 -5.040501e-07 -6.187050e-07 -3.996449e-07 -1.520050e-06
## [17,] -1.698550e-06 -8.594748e-07 -1.176841e-06 -7.766390e-07 -1.597800e-06
## [18,] -7.182063e-08 -1.870809e-08 -5.463117e-08 -4.327813e-08 -3.028556e-08
## [19,]  2.479168e-08  1.740528e-08  5.038281e-08  4.200075e-08  5.070093e-08
## [20,] -8.412429e-09  4.805712e-09 -3.181646e-09 -4.450111e-09  1.088428e-08
## [21,]  1.105869e-07 -9.767305e-09 -4.769298e-08 -3.685754e-08 -1.900740e-08
## [22,]  1.733709e-07 -4.357470e-08 -2.321355e-08 -2.399703e-10  6.592876e-09
## [23,] -3.272623e-08 -1.736834e-07 -4.130813e-08  5.619199e-09  1.586510e-08
## [24,] -7.152441e-08 -5.149901e-08  3.155614e-07  1.410936e-07 -2.242083e-08
## [25,] -3.399790e-08  7.772968e-11  1.598568e-07  1.585318e-07  2.066411e-08
## [26,]  3.556279e-08  2.884986e-08  4.141242e-08  6.360896e-08 -5.300243e-08
## [27,]  4.843826e-08  3.254177e-08  4.746746e-08  3.420047e-08 -6.824914e-08
## [28,] -3.075804e-08  7.837583e-09 -4.677204e-08 -4.646579e-08 -4.843254e-08
## [29,]  1.557847e-08  3.936364e-09  1.674782e-08  1.418938e-08  1.731460e-08
## [30,]  1.755251e-07  8.258053e-08  1.043056e-07  6.302207e-08  1.144535e-07
## [31,]  9.982544e-08  6.480777e-08  7.048046e-08  3.882008e-08  9.724019e-08
## [32,]  4.574408e-08  3.700744e-08  4.909661e-08  3.103637e-08  5.464245e-08
## [33,] -4.400080e-07  2.055843e-08 -8.151899e-09 -3.028220e-08  3.104888e-08
## [34,] -8.385078e-07 -4.893843e-07 -8.283728e-08 -3.214399e-08  1.347222e-08
## [35,] -5.306391e-07 -7.171173e-07 -2.813233e-07 -5.765006e-08  6.762329e-09
## [36,] -4.935546e-08 -2.292959e-07 -1.044457e-06 -3.581262e-07  1.729536e-08
## [37,]  2.034854e-08 -6.139984e-09 -3.355641e-07 -2.736634e-07 -1.291621e-07
## [38,]  7.236140e-08  6.052649e-08  5.925803e-08 -1.117344e-07 -6.461731e-07
## [39,]  7.054240e-08  5.701671e-08  7.268306e-08  2.981553e-08 -3.099615e-07
## [40,]  1.611696e-08  3.950314e-08 -2.460865e-08 -4.478314e-08 -1.050269e-07
## [41,] -7.937082e-09 -6.723492e-09  3.536067e-09  1.028254e-08  4.031094e-08
## [42,] -4.828625e-07 -1.604329e-07 -3.245144e-07 -2.370381e-07 -2.383833e-07
## [43,]  9.461212e-08  7.349475e-08  2.115852e-07  1.750903e-07  2.093632e-07
## [44,] -3.095669e-08  1.471234e-08 -3.661455e-09 -8.066384e-09  3.964872e-08
## [45,]  3.854305e-07 -6.169216e-08 -1.074642e-07 -6.921501e-08 -5.029490e-08
## [46,]  4.609527e-07 -1.599959e-07 -4.379038e-08  7.614437e-09  1.453747e-08
## [47,] -1.399748e-07 -3.626144e-07 -1.512954e-07 -5.047201e-08  1.703735e-08
## [48,] -8.000707e-08 -9.988733e-08 -2.459030e-07 -5.082716e-08 -1.872062e-09
## [49,] -1.440266e-08  2.729362e-09 -6.946297e-09  1.894328e-08 -1.179482e-07
## [50,]  1.418476e-07  1.190451e-07  1.340431e-07 -4.289829e-08 -8.234870e-07
## [51,]  1.871179e-07  1.351514e-07  1.639790e-07  8.418713e-08 -3.737835e-07
## [52,] -9.120175e-08  2.134079e-08 -1.126813e-07 -1.189492e-07 -2.045003e-07
## [53,]  5.656584e-08  1.691555e-08  4.593442e-08  4.079599e-08  9.049473e-08
## [54,]  1.382170e-06  5.706166e-07  9.097605e-07  6.159209e-07  8.507702e-07
## [55,]  5.904268e-07  3.370151e-07  4.634472e-07  2.965651e-07  5.510101e-07
## [56,]  1.714829e-07  1.269990e-07  1.554324e-07  9.427360e-08  1.872756e-07
## [57,] -1.273181e-06  1.189169e-07 -1.687557e-07 -2.337975e-07  2.203266e-08
## [58,] -1.992285e-06 -1.002151e-06 -2.956831e-07 -2.035137e-07 -1.916285e-08
## [59,] -1.149911e-06 -1.396583e-06 -4.666268e-07 -1.623605e-07  3.285308e-09
## [60,] -1.513434e-07 -3.139658e-07 -1.624495e-06 -5.618642e-07  1.720695e-08
## [61,]  1.001888e-08 -4.279685e-09 -4.881638e-07 -3.024492e-07 -2.900360e-07
## [62,]  2.937690e-07  2.454011e-07  2.658591e-07 -1.575453e-07 -1.871266e-06
## [63,]  2.270473e-07  2.190376e-07  2.298862e-07  9.376540e-08 -9.292369e-07
## [64,]  2.132269e-07  1.846034e-07  5.532938e-08 -3.538897e-08 -2.251384e-07
## [65,] -1.137058e-07 -5.224511e-08 -6.269253e-08 -2.645135e-08  7.301313e-08
## [66,] -1.526080e-05 -4.299249e-06 -2.218816e-05 -1.939079e-05 -1.241507e-05
##               [,11]         [,12]         [,13]         [,14]         [,15]
##  [1,] -2.984415e-07 -1.676348e-06  3.837901e-07 -8.551570e-07 -1.289480e-05
##  [2,]  1.809009e-07 -3.308032e-07  2.072744e-07 -6.831185e-08 -7.653976e-07
##  [3,]  4.782518e-08  1.673559e-08  5.203185e-09  2.558705e-07 -9.777198e-07
##  [4,]  6.883375e-08  2.449224e-08  2.883411e-09 -3.131077e-07 -9.141577e-07
##  [5,]  1.173174e-07  3.773776e-08  5.111411e-09 -8.770093e-08 -1.632436e-06
##  [6,]  7.150613e-08  2.739159e-08  1.895142e-09  9.149868e-08 -1.413283e-06
##  [7,]  3.074398e-08  2.264042e-08 -1.451909e-09 -2.122490e-07 -1.072105e-06
##  [8,]  9.557878e-08  2.861010e-08  5.527929e-09  3.379792e-07 -1.546715e-06
##  [9,]  5.271740e-08  9.967844e-09  1.086818e-08  2.604244e-07 -8.664735e-07
## [10,] -3.728258e-07 -1.258520e-07  4.179298e-08 -1.477561e-07 -8.539284e-07
## [11,]  3.993314e-04 -7.433355e-08 -1.332662e-07 -4.034124e-07 -6.337085e-07
## [12,] -7.433355e-08  3.993620e-04  1.350380e-07 -5.446653e-07 -1.024964e-06
## [13,] -1.332662e-07  1.350380e-07  3.996269e-04 -6.692764e-08  1.364628e-07
## [14,] -4.034124e-07 -5.446653e-07 -6.692764e-08  3.854309e-04 -5.794746e-07
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##               [,16]         [,17]         [,18]         [,19]         [,20]
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## [27,] -2.551129e-06 -3.079801e-06  9.326881e-08  7.156066e-08  2.494567e-08
## [28,] -2.918888e-06 -7.076084e-07 -2.490361e-07  2.347962e-09  3.883211e-08
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## [50,] -1.182860e-05 -1.125308e-05 -6.451773e-08  2.440870e-07  1.214071e-07
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## [52,] -1.119342e-05 -9.010960e-07 -6.592187e-07  1.157292e-08  1.289198e-07
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## [60,] -4.279987e-06 -3.365368e-06  4.264741e-08  2.007096e-08  6.041758e-08
## [61,] -2.815932e-06 -2.423009e-06  7.057593e-09  3.425739e-08  3.433293e-08
## [62,] -2.404563e-05 -2.317509e-05 -1.407621e-07  4.931705e-07  2.453924e-07
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## [65,] -4.255208e-06  7.821312e-06 -1.506008e-07 -1.051856e-07  3.257159e-08
## [66,]  9.057870e-05  1.570613e-04  1.018219e-05 -1.238544e-05  1.297754e-06
##               [,21]         [,22]         [,23]         [,24]         [,25]
##  [1,] -1.575422e-07  2.134833e-07 -1.467417e-07  1.795823e-07  2.112884e-07
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##  [4,] -1.064912e-07 -7.477695e-09  5.032730e-09 -4.728880e-08 -3.167199e-08
##  [5,]  6.655866e-08  1.454313e-07  1.329346e-08 -8.927159e-08 -5.811371e-08
##  [6,]  1.105869e-07  1.733709e-07 -3.272623e-08 -7.152441e-08 -3.399790e-08
##  [7,] -9.767305e-09 -4.357470e-08 -1.736834e-07 -5.149901e-08  7.772968e-11
##  [8,] -4.769298e-08 -2.321355e-08 -4.130813e-08  3.155614e-07  1.598568e-07
##  [9,] -3.685754e-08 -2.399703e-10  5.619199e-09  1.410936e-07  1.585318e-07
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## [16,] -4.472386e-06 -3.196252e-06 -1.959613e-06 -1.288802e-06 -5.288036e-07
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## [27,] -1.384945e-08  8.332051e-09  3.491080e-08  1.167103e-08 -1.480612e-08
## [28,]  6.549618e-08  3.038655e-08  4.013070e-08  8.436472e-08  4.339747e-08
## [29,] -2.400122e-08 -1.106234e-08 -5.213547e-09 -1.636407e-08 -5.742619e-09
## [30,] -3.282279e-07 -1.789428e-07  6.765372e-09 -8.902796e-09 -4.845001e-09
## [31,] -1.202379e-07 -4.874210e-08  4.575852e-08  3.249113e-08  1.475018e-08
## [32,] -3.683897e-08 -3.982595e-08  2.150809e-08  5.190066e-09  1.091179e-10
## [33,]  5.845736e-07  4.599435e-07  3.645337e-08  6.774783e-08  4.341583e-08
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## [36,] -2.214067e-09 -4.215087e-08 -8.887679e-08 -1.446339e-07  3.848434e-08
## [37,] -6.260458e-09  1.409892e-09  6.118771e-09  2.365474e-08  4.910468e-08
## [38,]  3.472107e-09  2.650668e-08  7.549720e-08  5.668826e-08 -9.726716e-08
## [39,]  1.954188e-08  4.013262e-08  7.524979e-08  4.088640e-08 -2.883959e-09
## [40,] -8.727698e-09 -1.153946e-08  5.863491e-08  1.104750e-07  5.819638e-08
## [41,]  2.549935e-08  1.787629e-08 -2.904465e-09 -1.351930e-08 -3.522782e-09
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## [43,]  2.591014e-07  2.760873e-07  1.571013e-07 -3.063758e-08 -4.515591e-08
## [44,] -1.138275e-06  5.632010e-08  7.764668e-08  7.237570e-08  3.454617e-08
## [45,] -3.807583e-06 -1.281143e-06  2.878320e-09  1.034829e-07  5.274413e-08
## [46,] -1.351409e-06 -1.952337e-06 -1.090920e-06 -1.025784e-07  1.006244e-08
## [47,] -1.587762e-08 -1.114289e-06 -1.411176e-06 -2.543826e-07  2.519011e-08
## [48,]  6.485841e-08 -1.073494e-07 -3.058899e-07 -1.451551e-06 -4.156326e-07
## [49,]  2.982450e-08  1.422796e-08 -4.392020e-09 -4.283067e-07 -2.531042e-07
## [50,]  1.313831e-08  5.928526e-08  1.557224e-07  1.079075e-07 -2.689377e-07
## [51,] -6.674219e-08  1.857110e-08  1.495601e-07  8.612595e-08 -6.935261e-09
## [52,]  2.462364e-07  1.312521e-07  1.250464e-07  2.254942e-07  1.152495e-07
## [53,] -1.092281e-07 -5.738779e-08 -1.602544e-08 -3.920547e-08 -1.161922e-08
## [54,] -2.596218e-06 -1.381205e-06 -6.243374e-08 -3.324771e-07 -1.895544e-07
## [55,] -6.546237e-07 -2.594248e-07  2.163309e-07  5.188859e-08  5.224053e-09
## [56,]  3.332664e-07 -1.207121e-07  7.874036e-08  3.409923e-08  1.058333e-08
## [57,]  3.222869e-06  1.863307e-06  8.073639e-08  3.376781e-07  2.228266e-07
## [58,]  2.161859e-06  1.537670e-06 -4.690461e-07  1.719163e-07  1.839569e-07
## [59,]  8.547662e-08 -5.457491e-07 -1.079644e-06 -9.732661e-08  1.035881e-07
## [60,]  4.712432e-08 -1.083866e-07 -2.511051e-07 -1.161144e-06 -2.143641e-07
## [61,]  1.956202e-08  1.261169e-08  1.395958e-08 -2.387025e-07 -1.273888e-07
## [62,]  1.758668e-08  1.143902e-07  3.162738e-07  2.277799e-07 -4.363286e-07
## [63,]  1.741404e-07  2.063853e-07  3.301654e-07  2.360302e-07  3.537076e-08
## [64,] -2.830047e-07 -1.538405e-07  1.829941e-07  2.776452e-07  1.375789e-07
## [65,]  2.265220e-07  1.261149e-07 -1.009417e-09  4.000372e-09  2.015995e-08
## [66,]  7.913297e-06 -1.554400e-06  9.980407e-07  1.880316e-05  1.202925e-05
##               [,26]         [,27]         [,28]         [,29]         [,30]
##  [1,]  2.304404e-07  1.415656e-07 -8.426327e-07  6.627573e-08 -2.061934e-08
##  [2,]  3.658512e-08  7.250357e-08 -6.208568e-08  4.841503e-08 -6.363355e-07
##  [3,]  3.793090e-08  4.695195e-08 -1.043983e-09  1.017957e-08 -1.688978e-07
##  [4,]  3.331888e-08  4.325408e-08 -2.613577e-08  1.242478e-08  7.497963e-08
##  [5,]  5.259152e-08  7.282597e-08 -5.767323e-08  2.508919e-08  2.556779e-07
##  [6,]  3.556279e-08  4.843826e-08 -3.075804e-08  1.557847e-08  1.755251e-07
##  [7,]  2.884986e-08  3.254177e-08  7.837583e-09  3.936364e-09  8.258053e-08
##  [8,]  4.141242e-08  4.746746e-08 -4.677204e-08  1.674782e-08  1.043056e-07
##  [9,]  6.360896e-08  3.420047e-08 -4.646579e-08  1.418938e-08  6.302207e-08
## [10,] -5.300243e-08 -6.824914e-08 -4.843254e-08  1.731460e-08  1.144535e-07
## [11,] -7.382049e-08 -2.594696e-07 -8.236903e-08 -3.671565e-08  2.172423e-07
## [12,] -1.417487e-09 -9.930537e-09 -2.451688e-07  1.560114e-09 -1.628440e-07
## [13,]  6.165407e-09 -5.158155e-08 -8.277902e-09 -2.166817e-07  1.372832e-07
## [14,] -9.836703e-07 -6.017211e-07 -1.196068e-06  2.076720e-07  1.847612e-07
## [15,] -6.433304e-07 -6.802591e-07 -7.068608e-07 -5.368592e-08 -8.153230e-07
## [16,] -2.971297e-06 -2.551129e-06 -2.918888e-06  5.769754e-07  5.807124e-06
## [17,] -2.719219e-06 -3.079801e-06 -7.076084e-07 -4.019227e-07 -1.158867e-05
## [18,] -1.354727e-08  9.326881e-08 -2.490361e-07  1.563958e-07  3.766824e-07
## [19,]  6.520777e-08  7.156066e-08  2.347962e-09  1.083509e-08  1.834484e-07
## [20,]  3.104110e-08  2.494567e-08  3.883211e-08 -7.431937e-09 -4.702023e-08
## [21,]  6.352694e-09 -1.384945e-08  6.549618e-08 -2.400122e-08 -3.282279e-07
## [22,]  1.830300e-08  8.332051e-09  3.038655e-08 -1.106234e-08 -1.789428e-07
## [23,]  3.928190e-08  3.491080e-08  4.013070e-08 -5.213547e-09  6.765372e-09
## [24,]  1.677520e-08  1.167103e-08  8.436472e-08 -1.636407e-08 -8.902796e-09
## [25,] -1.419510e-07 -1.480612e-08  4.339747e-08 -5.742619e-09 -4.845001e-09
## [26,]  3.993383e-04 -3.242785e-07 -8.870097e-08  4.689661e-08  1.771978e-07
## [27,] -3.242785e-07  3.993818e-04 -7.428807e-08 -1.265505e-07  2.347596e-07
## [28,] -8.870097e-08 -7.428807e-08  3.995016e-04  9.079729e-08 -2.015001e-08
## [29,]  4.689661e-08 -1.265505e-07  9.079729e-08  3.996655e-04  6.974033e-08
## [30,]  1.771978e-07  2.347596e-07 -2.015001e-08  6.974033e-08  3.976545e-04
## [31,]  1.671816e-07  1.755682e-07  1.015006e-07  5.084521e-09 -5.532032e-07
## [32,]  8.098565e-08  9.133988e-08  2.963941e-08  1.148777e-08  2.005857e-07
## [33,]  1.007527e-07  1.198156e-07  7.565070e-08  1.824869e-08  7.847681e-07
## [34,]  6.509734e-08  7.418824e-08  6.776052e-08  7.175647e-09  4.710374e-07
## [35,]  5.812065e-08  5.359949e-08  8.828622e-08 -8.039615e-09  1.585255e-07
## [36,]  5.739879e-08  5.202105e-08  5.368752e-08 -2.412441e-09  1.243155e-07
## [37,] -9.027286e-08  1.306455e-08  1.055511e-08  6.768411e-09  7.096106e-08
## [38,] -8.561429e-07 -4.296435e-07 -9.723588e-08  6.823894e-08  3.757705e-07
## [39,] -4.398156e-07 -9.274066e-07 -2.361846e-07 -1.602376e-07  6.292289e-07
## [40,] -7.981326e-08 -2.111992e-07 -4.842609e-07  1.728804e-08 -3.858495e-07
## [41,]  4.646223e-08 -1.846222e-07  8.150131e-09 -3.503099e-07  3.380235e-07
## [42,] -2.419588e-07  1.543149e-08 -7.048319e-07  3.963935e-07  4.582385e-06
## [43,]  2.609097e-07  2.994428e-07 -1.718146e-08  6.122607e-08  1.269070e-06
## [44,]  1.035545e-07  8.526061e-08  1.191392e-07 -2.234553e-08 -2.247884e-07
## [45,] -1.709763e-08 -8.344514e-08  1.507858e-07 -7.714988e-08 -1.349838e-06
## [46,]  3.144548e-08  7.242513e-11  7.236154e-08 -3.508455e-08 -6.659833e-07
## [47,]  9.845610e-08  8.364031e-08  1.359310e-07 -1.944967e-08  4.441647e-08
## [48,]  8.031954e-08  6.638218e-08  1.411305e-07 -2.061334e-08  8.419529e-08
## [49,] -2.727206e-07 -3.699919e-09  5.753705e-08  2.475885e-09  6.609438e-08
## [50,] -1.944098e-06 -9.646572e-07 -2.224654e-07  1.599995e-07  7.576371e-07
## [51,] -9.429497e-07 -1.840622e-06 -3.924879e-07 -3.271565e-07  6.853742e-07
## [52,] -3.023435e-07 -4.928702e-07 -1.086537e-06  1.900719e-07  6.321451e-07
## [53,]  1.665569e-07 -3.216630e-07  1.954679e-07 -5.902105e-07 -1.602659e-07
## [54,]  1.151155e-06  1.101726e-06  7.519719e-07 -1.573042e-07 -1.500217e-05
## [55,]  8.683969e-07  8.638870e-07  5.588906e-07 -3.994178e-08 -3.334644e-06
## [56,]  2.970636e-07  3.337105e-07  1.151535e-07  4.060101e-08  7.297109e-07
## [57,]  3.226419e-07  3.894039e-07  3.152132e-07  6.078775e-08  3.065684e-06
## [58,]  2.004861e-07  2.248863e-07  2.740403e-07  1.530521e-08  1.684969e-06
## [59,]  1.672219e-07  1.500234e-07  2.677838e-07 -2.796077e-08  4.217451e-07
## [60,]  1.460341e-07  1.257446e-07  2.056288e-07 -2.246610e-08  2.706689e-07
## [61,] -4.546384e-07  6.225918e-09  7.687360e-08  1.336720e-08  1.796828e-07
## [62,] -3.638440e-06 -1.855416e-06 -4.105649e-07  3.240192e-07  1.572070e-06
## [63,] -1.908258e-06 -3.518740e-06 -1.078868e-06 -4.182323e-07  2.802106e-06
## [64,] -3.526062e-07 -1.014410e-06 -1.219132e-06  1.315142e-08 -2.062801e-06
## [65,]  1.772239e-07 -5.788089e-07 -3.265516e-08 -7.311689e-07  1.615359e-06
## [66,] -3.571892e-06 -6.210078e-06  1.501497e-05 -3.646624e-06 -8.113671e-06
##               [,31]         [,32]         [,33]         [,34]         [,35]
##  [1,] -6.933501e-07 -6.555488e-07 -1.679837e-06 -1.648317e-06 -1.488793e-06
##  [2,] -2.509352e-07  1.370992e-08  1.197064e-07  7.301378e-08  2.991577e-08
##  [3,] -9.713017e-07 -2.455097e-07  7.258005e-08  5.208042e-08  2.125771e-08
##  [4,] -2.224666e-07 -4.315758e-07 -4.056090e-07 -1.050214e-08 -1.252219e-08
##  [5,]  1.283729e-07 -3.437790e-07 -1.264772e-06 -4.871504e-07 -5.861372e-08
##  [6,]  9.982544e-08  4.574408e-08 -4.400080e-07 -8.385078e-07 -5.306391e-07
##  [7,]  6.480777e-08  3.700744e-08  2.055843e-08 -4.893843e-07 -7.171173e-07
##  [8,]  7.048046e-08  4.909661e-08 -8.151899e-09 -8.283728e-08 -2.813233e-07
##  [9,]  3.882008e-08  3.103637e-08 -3.028220e-08 -3.214399e-08 -5.765006e-08
## [10,]  9.724019e-08  5.464245e-08  3.104888e-08  1.347222e-08  6.762329e-09
## [11,]  9.168151e-08  5.259082e-08 -1.078186e-08 -2.093903e-08 -2.034931e-08
## [12,]  8.581286e-08  5.002684e-08  7.232058e-08  4.962202e-08  3.900340e-08
## [13,] -8.493245e-09 -6.532784e-09 -2.455242e-08 -1.812173e-08 -1.211695e-08
## [14,]  3.318583e-07 -1.234095e-07  1.134532e-06  9.066528e-07 -2.169467e-07
## [15,] -2.023100e-06 -1.640200e-06 -2.840231e-06 -2.253625e-06 -1.525419e-06
## [16,] -1.547125e-06 -7.231620e-07  4.482046e-06  2.445425e-06 -1.173710e-06
## [17,] -6.916259e-06 -3.180963e-06 -6.555536e-06 -3.870423e-06 -1.618253e-06
## [18,] -1.885821e-08 -1.040821e-07 -2.196557e-07 -1.209271e-07  2.812004e-09
## [19,]  3.012817e-08 -5.148456e-08 -1.092092e-07 -7.155106e-08 -6.499181e-09
## [20,] -5.025285e-08 -1.052054e-07  5.697047e-09 -1.905936e-08  1.996509e-08
## [21,] -1.202379e-07 -3.683897e-08  5.845736e-07  5.065132e-07  1.881502e-08
## [22,] -4.874210e-08 -3.982595e-08  4.599435e-07  4.562761e-07 -1.458803e-07
## [23,]  4.575852e-08  2.150809e-08  3.645337e-08 -1.120579e-07 -3.561534e-07
## [24,]  3.249113e-08  5.190066e-09  6.774783e-08  2.261572e-08 -3.985045e-08
## [25,]  1.475018e-08  1.091179e-10  4.341583e-08  3.881565e-08  3.680409e-08
## [26,]  1.671816e-07  8.098565e-08  1.007527e-07  6.509734e-08  5.812065e-08
## [27,]  1.755682e-07  9.133988e-08  1.198156e-07  7.418824e-08  5.359949e-08
## [28,]  1.015006e-07  2.963941e-08  7.565070e-08  6.776052e-08  8.828622e-08
## [29,]  5.084521e-09  1.148777e-08  1.824869e-08  7.175647e-09 -8.039615e-09
## [30,] -5.532032e-07  2.005857e-07  7.847681e-07  4.710374e-07  1.585255e-07
## [31,]  3.970446e-04 -6.272863e-07  4.071508e-07  2.652576e-07  1.398001e-07
## [32,] -6.272863e-07  3.989712e-04 -9.870679e-07  9.488640e-08  6.482000e-08
## [33,]  4.071508e-07 -9.870679e-07  3.966245e-04 -1.094423e-06  1.252363e-07
## [34,]  2.652576e-07  9.488640e-08 -1.094423e-06  3.982037e-04 -9.737865e-07
## [35,]  1.398001e-07  6.482000e-08  1.252363e-07 -9.737865e-07  3.986557e-04
## [36,]  1.148303e-07  5.539028e-08  1.150922e-07 -4.379936e-08 -2.885497e-07
## [37,]  6.175361e-08  3.158244e-08  4.435937e-08  3.084388e-08 -1.026496e-08
## [38,]  3.466484e-07  1.674063e-07  2.328815e-07  1.535253e-07  1.281391e-07
## [39,]  3.298837e-07  1.473197e-07  1.710071e-07  1.116314e-07  1.072839e-07
## [40,]  2.700970e-07  1.530567e-07  3.413789e-07  2.380978e-07  1.728419e-07
## [41,] -3.977533e-08 -3.605762e-08 -9.243255e-08 -6.048258e-08 -2.913078e-08
## [42,] -7.090517e-07 -9.415184e-07 -1.769404e-06 -1.019898e-06 -1.537320e-07
## [43,] -1.576014e-06 -6.200431e-07 -4.579247e-07 -3.201929e-07 -2.377332e-08
## [44,] -6.142248e-07 -7.225792e-08  4.341173e-07 -6.766188e-08  5.606348e-08
## [45,] -5.363055e-07  2.347521e-07  3.007323e-06  1.953928e-06 -6.313507e-08
## [46,] -2.173945e-07 -1.524919e-07  1.948921e-06  1.543758e-06 -6.046389e-07
## [47,]  1.362282e-07  5.694882e-08  1.907727e-07 -3.971395e-07 -1.049976e-06
## [48,]  1.363264e-07  5.398948e-08  1.514908e-07  1.082968e-08 -1.942263e-07
## [49,]  8.384394e-08  3.471451e-08  8.396524e-08  6.709524e-08  3.595323e-08
## [50,]  7.012991e-07  3.371883e-07  4.609375e-07  3.023660e-07  2.562230e-07
## [51,]  7.560210e-07  4.054438e-07  5.981705e-07  3.799469e-07  2.675851e-07
## [52,]  3.214569e-07  3.996381e-08  6.978354e-08  1.003813e-07  2.229294e-07
## [53,]  3.160191e-08  7.582209e-08  1.577205e-07  8.606206e-08  2.513920e-10
## [54,] -1.211958e-06  1.978365e-06  5.434308e-06  3.188967e-06  8.548339e-07
## [55,] -1.108035e-05 -1.970231e-06  2.077507e-06  1.261034e-06  5.989055e-07
## [56,] -2.367191e-06 -3.129192e-06 -3.194215e-06  3.500857e-07  2.420886e-07
## [57,]  1.559209e-06 -3.154680e-06 -1.199533e-05 -3.934757e-06  6.370627e-07
## [58,]  9.157635e-07  3.298074e-07 -4.119222e-06 -5.315420e-06 -2.250528e-06
## [59,]  3.867483e-07  1.737232e-07  3.339195e-07 -2.426388e-06 -3.275202e-06
## [60,]  2.872345e-07  1.255851e-07  3.289519e-07 -6.662100e-08 -5.378339e-07
## [61,]  1.791907e-07  8.185267e-08  1.659190e-07  1.246225e-07  3.319196e-08
## [62,]  1.445762e-06  6.949025e-07  9.620267e-07  6.312407e-07  5.282328e-07
## [63,]  1.372137e-06  5.560588e-07  6.436762e-07  4.401327e-07  4.794553e-07
## [64,]  1.076204e-06  7.440060e-07  1.534086e-06  1.004426e-06  5.893129e-07
## [65,] -2.147880e-07 -2.481778e-07 -5.314212e-07 -3.222827e-07 -1.062475e-07
## [66,] -3.140865e-06 -4.359057e-06  1.756284e-05  1.582217e-05  1.449993e-05
##               [,36]         [,37]         [,38]         [,39]         [,40]
##  [1,] -1.485382e-06 -6.110354e-07 -4.960866e-07 -6.117359e-08 -1.422390e-06
##  [2,]  2.932432e-08  1.763931e-08  7.916099e-08  1.846734e-07 -2.114870e-07
##  [3,]  2.714374e-08  1.867346e-08  8.089889e-08  7.660015e-08  4.939768e-08
##  [4,]  1.059529e-08  1.694071e-08  6.496812e-08  6.633759e-08  5.758143e-09
##  [5,]  1.462491e-08  2.823296e-08  1.036155e-07  1.045342e-07  4.609202e-09
##  [6,] -4.935546e-08  2.034854e-08  7.236140e-08  7.054240e-08  1.611696e-08
##  [7,] -2.292959e-07 -6.139984e-09  6.052649e-08  5.701671e-08  3.950314e-08
##  [8,] -1.044457e-06 -3.355641e-07  5.925803e-08  7.268306e-08 -2.460865e-08
##  [9,] -3.581262e-07 -2.736634e-07 -1.117344e-07  2.981553e-08 -4.478314e-08
## [10,]  1.729536e-08 -1.291621e-07 -6.461731e-07 -3.099615e-07 -1.050269e-07
## [11,]  1.180049e-08  4.704707e-09 -3.130160e-07 -6.021475e-07 -1.179738e-07
## [12,]  3.562465e-08  1.416302e-08 -4.771858e-08 -3.207234e-08 -5.097186e-07
## [13,] -6.956096e-09  2.324672e-09  3.824583e-08 -1.325381e-07  8.261071e-08
## [14,] -2.395415e-08  2.813712e-08 -1.063662e-06 -1.152495e-06 -9.642735e-07
## [15,] -1.467374e-06 -6.800692e-07 -1.577413e-06 -1.307782e-06 -1.631076e-06
## [16,] -1.347727e-06 -8.007801e-07 -5.711723e-06 -6.247480e-06 -2.684738e-06
## [17,] -1.389005e-06 -8.778874e-07 -5.475671e-06 -4.990344e-06 -4.528877e-06
## [18,] -4.543815e-09 -1.090199e-08 -3.321743e-08  1.621324e-08 -1.102434e-07
## [19,]  1.424805e-08  1.752179e-08  1.166221e-07  1.208133e-07  4.853672e-08
## [20,]  1.656867e-08  7.466427e-09  5.845881e-08  6.103419e-08  4.134035e-08
## [21,] -2.214067e-09 -6.260458e-09  3.472107e-09  1.954188e-08 -8.727698e-09
## [22,] -4.215087e-08  1.409892e-09  2.650668e-08  4.013262e-08 -1.153946e-08
## [23,] -8.887679e-08  6.118771e-09  7.549720e-08  7.524979e-08  5.863491e-08
## [24,] -1.446339e-07  2.365474e-08  5.668826e-08  4.088640e-08  1.104750e-07
## [25,]  3.848434e-08  4.910468e-08 -9.726716e-08 -2.883959e-09  5.819638e-08
## [26,]  5.739879e-08 -9.027286e-08 -8.561429e-07 -4.398156e-07 -7.981326e-08
## [27,]  5.202105e-08  1.306455e-08 -4.296435e-07 -9.274066e-07 -2.111992e-07
## [28,]  5.368752e-08  1.055511e-08 -9.723588e-08 -2.361846e-07 -4.842609e-07
## [29,] -2.412441e-09  6.768411e-09  6.823894e-08 -1.602376e-07  1.728804e-08
## [30,]  1.243155e-07  7.096106e-08  3.757705e-07  6.292289e-07 -3.858495e-07
## [31,]  1.148303e-07  6.175361e-08  3.466484e-07  3.298837e-07  2.700970e-07
## [32,]  5.539028e-08  3.158244e-08  1.674063e-07  1.473197e-07  1.530567e-07
## [33,]  1.150922e-07  4.435937e-08  2.328815e-07  1.710071e-07  3.413789e-07
## [34,] -4.379936e-08  3.084388e-08  1.535253e-07  1.116314e-07  2.380978e-07
## [35,] -2.885497e-07 -1.026496e-08  1.281391e-07  1.072839e-07  1.728419e-07
## [36,]  3.984860e-04 -4.614514e-07  1.120362e-07  1.004800e-07  1.165032e-07
## [37,] -4.614514e-07  3.997274e-04 -2.555339e-07  1.573806e-08  3.463997e-08
## [38,]  1.120362e-07 -2.555339e-07  3.980494e-04 -9.956170e-07 -1.781836e-07
## [39,]  1.004800e-07  1.573806e-08 -9.956170e-07  3.980848e-04 -3.816855e-07
## [40,]  1.165032e-07  3.463997e-08 -1.781836e-07 -3.816855e-07  3.989888e-04
## [41,] -1.844407e-08  3.940241e-09  1.124514e-07 -3.711052e-07  1.588412e-07
## [42,] -1.432792e-07 -1.162216e-07 -5.269711e-07 -6.223709e-07  3.212853e-08
## [43,]  6.420898e-08  7.355391e-08  4.709235e-07  4.284996e-07  3.400830e-07
## [44,]  5.082140e-08  2.481586e-08  1.944355e-07  2.029401e-07  1.316506e-07
## [45,] -6.091056e-08 -3.297156e-08 -7.613619e-08  4.790520e-09 -1.953126e-07
## [46,] -1.436087e-07 -2.748249e-09  3.167120e-08  8.190687e-08 -9.748940e-08
## [47,] -2.511895e-07 -3.018218e-09  1.972730e-07  1.864481e-07  2.038998e-07
## [48,] -1.243673e-06 -2.844281e-07  1.751465e-07  1.499137e-07  2.206786e-07
## [49,] -2.656168e-07 -1.509667e-07 -4.434107e-07  1.156090e-08  9.685374e-08
## [50,]  2.318706e-07 -4.118071e-07 -3.663991e-06 -1.929661e-06 -3.285823e-07
## [51,]  2.381560e-07  6.438270e-08 -1.837429e-06 -3.495043e-06 -9.530332e-07
## [52,]  1.426896e-07  2.571688e-08 -5.781930e-07 -1.266542e-06 -1.309987e-06
## [53,]  6.713115e-09  2.994392e-08  3.385397e-07 -4.089774e-07  3.493619e-08
## [54,]  7.228871e-07  4.696554e-07  2.425151e-06  3.602710e-06 -1.548019e-06
## [55,]  5.291676e-07  3.142671e-07  1.772533e-06  1.808420e-06  1.009371e-06
## [56,]  2.000209e-07  1.119440e-07  6.125177e-07  5.355741e-07  5.770584e-07
## [57,]  4.391676e-07  1.425603e-07  7.785550e-07  5.182179e-07  1.350364e-06
## [58,] -1.939050e-08  8.981100e-08  4.882032e-07  3.265015e-07  8.863995e-07
## [59,] -5.721107e-07 -1.566859e-08  3.651781e-07  3.065681e-07  5.034411e-07
## [60,] -3.323526e-06 -9.823947e-07  3.054907e-07  2.611856e-07  3.724199e-07
## [61,] -9.543276e-07 -5.712252e-07 -8.968506e-07  1.927105e-08  1.550261e-07
## [62,]  4.706629e-07 -8.098139e-07 -7.530731e-06 -4.122257e-06 -7.109106e-07
## [63,]  4.262831e-07  1.019899e-07 -4.139729e-06 -7.424245e-06 -1.736432e-06
## [64,]  4.235059e-07  1.614874e-07 -7.535748e-07 -1.807791e-06 -2.889274e-06
## [65,] -8.135638e-08 -8.715509e-09  3.990821e-07 -1.170787e-06  3.912310e-07
## [66,]  4.451148e-06 -2.014438e-06 -4.765784e-06 -8.077459e-06  1.737217e-05
##               [,41]         [,42]         [,43]         [,44]         [,45]
##  [1,]  2.121388e-07 -3.385292e-06  1.595862e-06 -6.455058e-08 -3.812199e-08
##  [2,]  1.470179e-07  2.224860e-07  2.573059e-07 -3.008161e-09 -1.882367e-07
##  [3,] -9.387142e-09 -3.589126e-07  6.559765e-08 -5.889163e-08 -1.553674e-07
##  [4,] -3.158845e-09 -3.513635e-07  8.918269e-08 -1.107162e-07 -9.357741e-08
##  [5,] -8.085157e-09 -7.246068e-07  1.811941e-07 -6.295850e-09  4.525079e-07
##  [6,] -7.937082e-09 -4.828625e-07  9.461212e-08 -3.095669e-08  3.854305e-07
##  [7,] -6.723492e-09 -1.604329e-07  7.349475e-08  1.471234e-08 -6.169216e-08
##  [8,]  3.536067e-09 -3.245144e-07  2.115852e-07 -3.661455e-09 -1.074642e-07
##  [9,]  1.028254e-08 -2.370381e-07  1.750903e-07 -8.066384e-09 -6.921501e-08
## [10,]  4.031094e-08 -2.383833e-07  2.093632e-07  3.964872e-08 -5.029490e-08
## [11,] -1.281470e-07 -2.170094e-07  2.770262e-07  3.814294e-08 -4.112565e-08
## [12,]  8.836376e-08 -2.793204e-07  1.252991e-07  4.644841e-08 -3.526280e-08
## [13,] -3.316838e-07  1.046124e-07  1.281636e-08 -1.531032e-09  1.801610e-08
## [14,] -9.967221e-08 -3.254780e-06 -4.418504e-06 -2.442520e-06 -4.938748e-06
## [15,]  2.000906e-07 -1.536366e-06 -5.735761e-07  1.358640e-07  2.612220e-06
## [16,] -4.859037e-07 -2.537992e-05 -1.991351e-05 -8.153056e-06 -1.634376e-05
## [17,]  1.067277e-06  2.662227e-05 -1.894762e-06 -3.413739e-07  9.751620e-06
## [18,]  7.642330e-08 -2.794083e-06 -1.453744e-06 -6.292478e-08  5.714363e-07
## [19,] -2.909645e-09 -1.363641e-06 -3.391793e-06 -7.975880e-07  2.223115e-07
## [20,]  1.784325e-09  4.344068e-08 -7.541425e-07 -1.110424e-06 -1.138912e-06
## [21,]  2.549935e-08  1.023137e-06  2.591014e-07 -1.138275e-06 -3.807583e-06
## [22,]  1.787629e-08  5.601167e-07  2.760873e-07  5.632010e-08 -1.281143e-06
## [23,] -2.904465e-09  1.128416e-07  1.571013e-07  7.764668e-08  2.878320e-09
## [24,] -1.351930e-08  2.286576e-07 -3.063758e-08  7.237570e-08  1.034829e-07
## [25,] -3.522782e-09  1.246722e-07 -4.515591e-08  3.454617e-08  5.274413e-08
## [26,]  4.646223e-08 -2.419588e-07  2.609097e-07  1.035545e-07 -1.709763e-08
## [27,] -1.846222e-07  1.543149e-08  2.994428e-07  8.526061e-08 -8.344514e-08
## [28,]  8.150131e-09 -7.048319e-07 -1.718146e-08  1.191392e-07  1.507858e-07
## [29,] -3.503099e-07  3.963935e-07  6.122607e-08 -2.234553e-08 -7.714988e-08
## [30,]  3.380235e-07  4.582385e-06  1.269070e-06 -2.247884e-07 -1.349838e-06
## [31,] -3.977533e-08 -7.090517e-07 -1.576014e-06 -6.142248e-07 -5.363055e-07
## [32,] -3.605762e-08 -9.415184e-07 -6.200431e-07 -7.225792e-08  2.347521e-07
## [33,] -9.243255e-08 -1.769404e-06 -4.579247e-07  4.341173e-07  3.007323e-06
## [34,] -6.048258e-08 -1.019898e-06 -3.201929e-07 -6.766188e-08  1.953928e-06
## [35,] -2.913078e-08 -1.537320e-07 -2.377332e-08  5.606348e-08 -6.313507e-08
## [36,] -1.844407e-08 -1.432792e-07  6.420898e-08  5.082140e-08 -6.091056e-08
## [37,]  3.940241e-09 -1.162216e-07  7.355391e-08  2.481586e-08 -3.297156e-08
## [38,]  1.124514e-07 -5.269711e-07  4.709235e-07  1.944355e-07 -7.613619e-08
## [39,] -3.711052e-07 -6.223709e-07  4.284996e-07  2.029401e-07  4.790520e-09
## [40,]  1.588412e-07  3.212853e-08  3.400830e-07  1.316506e-07 -1.953126e-07
## [41,]  3.994187e-04 -8.514268e-08 -9.344919e-08  4.612755e-09  1.232218e-07
## [42,] -8.514268e-08  3.822979e-04 -7.591749e-06 -7.820667e-08  3.891695e-06
## [43,] -9.344919e-08 -7.591749e-06  3.864774e-04 -2.904558e-06  1.101948e-06
## [44,]  4.612755e-09 -7.820667e-08 -2.904558e-06  3.965626e-04 -3.743839e-06
## [45,]  1.232218e-07  3.891695e-06  1.101948e-06 -3.743839e-06  3.862803e-04
## [46,]  6.864754e-08  1.961088e-06  8.791247e-07  1.454904e-07 -4.965934e-06
## [47,] -1.899201e-08  2.886520e-07  2.722880e-07  1.930836e-07 -8.677074e-08
## [48,] -2.800075e-08  1.733149e-07  9.412374e-08  1.463500e-07  1.055324e-07
## [49,]  4.674788e-09  2.804658e-08  6.547894e-08  7.492268e-08  4.260763e-08
## [50,]  2.274672e-07 -1.056357e-06  9.808965e-07  4.046645e-07 -1.325754e-07
## [51,] -5.259622e-07  2.443654e-07  1.277702e-06  3.583863e-07 -4.316950e-07
## [52,] -1.244180e-08 -3.558722e-06 -3.149636e-07  3.969837e-07  6.922408e-07
## [53,] -7.354975e-07  1.946680e-06  3.815724e-07 -8.374967e-08 -3.968751e-07
## [54,]  1.501045e-06  4.042431e-05  9.832291e-06 -1.926610e-06 -1.028764e-05
## [55,] -4.665577e-08  6.822682e-07 -1.042267e-05 -3.787341e-06 -2.828617e-06
## [56,] -1.380047e-07 -3.923758e-06 -3.969371e-06  4.333893e-07  2.149683e-06
## [57,] -3.780203e-07 -6.986697e-06 -2.287844e-06  2.666710e-06  1.511125e-05
## [58,] -2.287294e-07 -3.635719e-06 -1.443921e-06 -2.025274e-07  8.682975e-06
## [59,] -8.278386e-08 -3.320289e-07 -6.277747e-08  1.835398e-07 -4.109897e-08
## [60,] -5.567269e-08 -9.991132e-08  9.280818e-08  1.831940e-07 -3.930105e-08
## [61,]  1.122520e-08 -1.553443e-07  1.426685e-07  1.083333e-07 -1.968500e-08
## [62,]  5.095831e-07 -2.218988e-06  1.982834e-06  8.183503e-07 -3.068736e-07
## [63,] -1.052408e-06 -4.323876e-06  1.346055e-06  9.164065e-07  3.164284e-07
## [64,]  4.231532e-07  4.207943e-06  2.410581e-06  3.256870e-07 -1.576356e-06
## [65,] -1.454784e-06 -2.775501e-06 -1.030463e-06  8.559059e-08  9.297150e-07
## [66,] -3.046768e-06  5.024874e-05 -4.958148e-05  2.047514e-06  9.200804e-06
##               [,46]         [,47]         [,48]         [,49]         [,50]
##  [1,]  5.648680e-07 -5.661182e-07 -4.664659e-07 -9.888857e-08 -7.405878e-08
##  [2,] -8.474648e-08  1.430628e-08  1.326195e-08  1.076593e-08  1.558495e-07
##  [3,] -5.261757e-08  1.169984e-08  3.605053e-09  6.293541e-09  1.591564e-07
##  [4,] -3.402557e-08 -1.762835e-08 -3.950586e-08 -1.424979e-08  1.286055e-07
##  [5,]  5.002210e-07 -1.592823e-08 -8.093377e-08 -3.045218e-08  2.046428e-07
##  [6,]  4.609527e-07 -1.399748e-07 -8.000707e-08 -1.440266e-08  1.418476e-07
##  [7,] -1.599959e-07 -3.626144e-07 -9.988733e-08  2.729362e-09  1.190451e-07
##  [8,] -4.379038e-08 -1.512954e-07 -2.459030e-07 -6.946297e-09  1.340431e-07
##  [9,]  7.614437e-09 -5.047201e-08 -5.082716e-08  1.894328e-08 -4.289829e-08
## [10,]  1.453747e-08  1.703735e-08 -1.872062e-09 -1.179482e-07 -8.234870e-07
## [11,]  3.725663e-08 -5.636272e-09 -4.042478e-08 -3.165773e-08 -4.032132e-07
## [12,] -2.149155e-09  4.927078e-08  4.159548e-08  1.875572e-08 -2.051708e-08
## [13,]  1.422857e-08 -9.861763e-09 -1.364624e-08 -1.443093e-09  5.187424e-08
## [14,] -3.717637e-06 -2.141211e-06 -1.685513e-06 -7.352686e-07 -2.509340e-06
## [15,]  1.498570e-06 -5.133265e-07 -5.007667e-07 -2.441193e-07 -2.716963e-06
## [16,] -1.021831e-05 -4.771502e-06 -3.286957e-06 -1.770091e-06 -1.182860e-05
## [17,]  4.035352e-06 -1.613704e-06 -1.561851e-06 -1.095286e-06 -1.125308e-05
## [18,]  2.899926e-07  7.871412e-08  6.293847e-08  2.352448e-08 -6.451773e-08
## [19,]  1.982306e-07  6.406386e-08  2.208698e-08  1.593752e-08  2.440870e-07
## [20,]  3.952873e-08  6.050796e-08  4.762152e-08  2.404978e-08  1.214071e-07
## [21,] -1.351409e-06 -1.587762e-08  6.485841e-08  2.982450e-08  1.313831e-08
## [22,] -1.952337e-06 -1.114289e-06 -1.073494e-07  1.422796e-08  5.928526e-08
## [23,] -1.090920e-06 -1.411176e-06 -3.058899e-07 -4.392020e-09  1.557224e-07
## [24,] -1.025784e-07 -2.543826e-07 -1.451551e-06 -4.283067e-07  1.079075e-07
## [25,]  1.006244e-08  2.519011e-08 -4.156326e-07 -2.531042e-07 -2.689377e-07
## [26,]  3.144548e-08  9.845610e-08  8.031954e-08 -2.727206e-07 -1.944098e-06
## [27,]  7.242513e-11  8.364031e-08  6.638218e-08 -3.699919e-09 -9.646572e-07
## [28,]  7.236154e-08  1.359310e-07  1.411305e-07  5.753705e-08 -2.224654e-07
## [29,] -3.508455e-08 -1.944967e-08 -2.061334e-08  2.475885e-09  1.599995e-07
## [30,] -6.659833e-07  4.441647e-08  8.419529e-08  6.609438e-08  7.576371e-07
## [31,] -2.173945e-07  1.362282e-07  1.363264e-07  8.384394e-08  7.012991e-07
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## [50,]  7.993605e-08  4.052993e-07  3.523305e-07 -8.769957e-07  3.924452e-04
## [51,] -6.344467e-08  3.857872e-07  3.312849e-07  5.488035e-08 -3.956297e-06
## [52,]  3.655148e-07  3.960385e-07  3.858658e-07  1.529807e-07 -1.235089e-06
## [53,] -1.958011e-07 -4.907719e-08 -4.168412e-08  2.391555e-08  7.371284e-07
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## [55,] -1.140181e-06  5.756856e-07  5.398736e-07  3.607704e-07  3.601265e-06
## [56,] -4.517727e-07  2.184847e-07  2.134939e-07  1.350074e-07  1.237830e-06
## [57,]  8.284322e-06  6.828654e-07  6.417429e-07  3.451225e-07  1.536720e-06
## [58,]  5.936469e-06 -1.499984e-06  1.371419e-07  2.638056e-07  9.634531e-07
## [59,] -2.129818e-06 -3.469751e-06 -5.902032e-07  9.950698e-08  7.354653e-07
## [60,] -4.150470e-07 -7.196111e-07 -4.237989e-06 -1.053039e-06  6.202978e-07
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## [62,]  1.420608e-07  8.261575e-07  7.239407e-07 -1.607439e-06 -1.515291e-05
## [63,]  4.685126e-07  8.487383e-07  7.039471e-07  1.452341e-07 -8.703126e-06
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## [66,] -7.464115e-06  1.460847e-05  2.248193e-05  9.805632e-06 -9.925606e-06
##               [,51]         [,52]         [,53]         [,54]         [,55]
##  [1,]  2.067333e-07 -2.037791e-06  3.406248e-07  6.652979e-06  5.872295e-07
##  [2,]  2.227522e-07 -7.381924e-08  8.501150e-08 -2.006045e-06 -6.770527e-07
##  [3,]  1.948957e-07 -9.530580e-09  4.121024e-08  3.072726e-07 -2.748992e-06
##  [4,]  1.625305e-07 -7.110718e-08  4.217995e-08  8.613631e-07 -3.448906e-07
##  [5,]  2.737586e-07 -1.640797e-07  8.734880e-08  2.060142e-06  8.681840e-07
##  [6,]  1.871179e-07 -9.120175e-08  5.656584e-08  1.382170e-06  5.904268e-07
##  [7,]  1.351514e-07  2.134079e-08  1.691555e-08  5.706166e-07  3.370151e-07
##  [8,]  1.639790e-07 -1.126813e-07  4.593442e-08  9.097605e-07  4.634472e-07
##  [9,]  8.418713e-08 -1.189492e-07  4.079599e-08  6.159209e-07  2.965651e-07
## [10,] -3.737835e-07 -2.045003e-07  9.049473e-08  8.507702e-07  5.510101e-07
## [11,] -8.617700e-07 -3.840319e-07 -1.328485e-07  1.154371e-06  5.753613e-07
## [12,] -1.276670e-07 -5.533531e-07  4.021466e-08  1.662452e-08  4.309812e-07
## [13,] -1.784669e-07  1.191914e-08 -3.522331e-07  2.677945e-07 -3.116697e-08
## [14,] -2.034879e-06 -2.644848e-06  4.757786e-07  3.137190e-06 -2.836289e-07
## [15,] -2.655874e-06 -2.159946e-06  2.323134e-08 -4.061235e-07 -6.546366e-06
## [16,] -1.030370e-05 -1.119342e-05  2.761807e-06  5.044270e-05 -6.277374e-06
## [17,] -1.357532e-05 -9.010960e-07 -2.495442e-06 -8.426531e-05 -3.788468e-05
## [18,]  2.142286e-07 -6.592187e-07  4.052194e-07  4.825500e-06  1.823939e-07
## [19,]  2.801647e-07  1.157292e-08  4.095682e-08  1.025353e-06 -1.513762e-06
## [20,]  1.050937e-07  1.289198e-07 -2.841969e-08 -5.464631e-07 -6.218474e-07
## [21,] -6.674219e-08  2.462364e-07 -1.092281e-07 -2.596218e-06 -6.546237e-07
## [22,]  1.857110e-08  1.312521e-07 -5.738779e-08 -1.381205e-06 -2.594248e-07
## [23,]  1.495601e-07  1.250464e-07 -1.602544e-08 -6.243374e-08  2.163309e-07
## [24,]  8.612595e-08  2.254942e-07 -3.920547e-08 -3.324771e-07  5.188859e-08
## [25,] -6.935261e-09  1.152495e-07 -1.161922e-08 -1.895544e-07  5.224053e-09
## [26,] -9.429497e-07 -3.023435e-07  1.665569e-07  1.151155e-06  8.683969e-07
## [27,] -1.840622e-06 -4.928702e-07 -3.216630e-07  1.101726e-06  8.638870e-07
## [28,] -3.924879e-07 -1.086537e-06  1.954679e-07  7.519719e-07  5.588906e-07
## [29,] -3.271565e-07  1.900719e-07 -5.902105e-07 -1.573042e-07 -3.994178e-08
## [30,]  6.853742e-07  6.321451e-07 -1.602659e-07 -1.500217e-05 -3.334644e-06
## [31,]  7.560210e-07  3.214569e-07  3.160191e-08 -1.211958e-06 -1.108035e-05
## [32,]  4.054438e-07  3.996381e-08  7.582209e-08  1.978365e-06 -1.970231e-06
## [33,]  5.981705e-07  6.978354e-08  1.577205e-07  5.434308e-06  2.077507e-06
## [34,]  3.799469e-07  1.003813e-07  8.606206e-08  3.188967e-06  1.261034e-06
## [35,]  2.675851e-07  2.229294e-07  2.513920e-10  8.548339e-07  5.989055e-07
## [36,]  2.381560e-07  1.426896e-07  6.713115e-09  7.228871e-07  5.291676e-07
## [37,]  6.438270e-08  2.571688e-08  2.994392e-08  4.696554e-07  3.142671e-07
## [38,] -1.837429e-06 -5.781930e-07  3.385397e-07  2.425151e-06  1.772533e-06
## [39,] -3.495043e-06 -1.266542e-06 -4.089774e-07  3.602710e-06  1.808420e-06
## [40,] -9.530332e-07 -1.309987e-06  3.493619e-08 -1.548019e-06  1.009371e-06
## [41,] -5.259622e-07 -1.244180e-08 -7.354975e-07  1.501045e-06 -4.665577e-08
## [42,]  2.443654e-07 -3.558722e-06  1.946680e-06  4.042431e-05  6.822682e-07
## [43,]  1.277702e-06 -3.149636e-07  3.815724e-07  9.832291e-06 -1.042267e-05
## [44,]  3.583863e-07  3.969837e-07 -8.374967e-08 -1.926610e-06 -3.787341e-06
## [45,] -4.316950e-07  6.922408e-07 -3.968751e-07 -1.028764e-05 -2.828617e-06
## [46,] -6.344467e-08  3.655148e-07 -1.958011e-07 -5.037124e-06 -1.140181e-06
## [47,]  3.857872e-07  3.960385e-07 -4.907719e-08 -8.671168e-08  5.756856e-07
## [48,]  3.312849e-07  3.858658e-07 -4.168412e-08  1.721916e-07  5.398736e-07
## [49,]  5.488035e-08  1.529807e-07  2.391555e-08  2.697005e-07  3.607704e-07
## [50,] -3.956297e-06 -1.235089e-06  7.371284e-07  4.898999e-06  3.601265e-06
## [51,]  3.929439e-04 -2.160261e-06 -8.504239e-07  3.038401e-06  3.542793e-06
## [52,] -2.160261e-06  3.965864e-04  6.021683e-07  7.105261e-06  2.282994e-06
## [53,] -8.504239e-07  6.021683e-07  3.984932e-04 -3.028856e-06 -3.198674e-07
## [54,]  3.038401e-06  7.105261e-06 -3.028856e-06  2.863521e-04 -1.690272e-05
## [55,]  3.542793e-06  2.282994e-06 -3.198674e-07 -1.690272e-05  3.532717e-04
## [56,]  1.492431e-06  1.520648e-07  2.824935e-07  7.133470e-06 -8.824615e-06
## [57,]  2.061324e-06  2.644276e-07  6.086507e-07  2.115522e-05  7.675919e-06
## [58,]  1.229349e-06  4.097710e-07  2.938259e-07  1.131217e-05  4.264992e-06
## [59,]  7.583240e-07  6.830317e-07 -1.409863e-08  2.186001e-06  1.637419e-06
## [60,]  6.123478e-07  5.462364e-07 -2.453321e-08  1.299628e-06  1.225352e-06
## [61,]  1.232347e-07  1.959638e-07  7.428220e-08  1.006008e-06  8.327324e-07
## [62,] -8.281730e-06 -2.576817e-06  1.578102e-06  1.018009e-05  7.418360e-06
## [63,] -1.425069e-05 -5.676894e-06 -7.184947e-07  1.811769e-05  7.931048e-06
## [64,] -4.521949e-06 -3.389323e-06 -3.433861e-07 -1.389849e-05  3.055636e-06
## [65,] -1.453069e-06 -5.865596e-07 -1.735594e-06  1.039450e-05  1.550522e-07
## [66,] -1.594710e-05  3.626311e-05 -8.524880e-06 -1.145423e-04 -4.958512e-05
##               [,56]         [,57]         [,58]         [,59]         [,60]
##  [1,] -1.225304e-06 -5.017787e-06 -4.220555e-06 -2.988305e-06 -2.408010e-06
##  [2,]  7.373955e-08  4.354222e-07  2.367368e-07  7.223578e-08  5.281503e-08
##  [3,] -6.446816e-07  2.709191e-07  1.470570e-07  4.567180e-08  3.811824e-08
##  [4,] -9.707759e-07 -1.173723e-06 -8.086236e-08 -5.569282e-08 -3.150115e-08
##  [5,] -8.946744e-07 -3.797297e-06 -1.500183e-06 -1.748094e-07 -6.805816e-08
##  [6,]  1.714829e-07 -1.273181e-06 -1.992285e-06 -1.149911e-06 -1.513434e-07
##  [7,]  1.269990e-07  1.189169e-07 -1.002151e-06 -1.396583e-06 -3.139658e-07
##  [8,]  1.554324e-07 -1.687557e-07 -2.956831e-07 -4.666268e-07 -1.624495e-06
##  [9,]  9.427360e-08 -2.337975e-07 -2.035137e-07 -1.623605e-07 -5.618642e-07
## [10,]  1.872756e-07  2.203266e-08 -1.916285e-08  3.285308e-09  1.720695e-08
## [11,]  1.772485e-07 -1.756896e-07 -1.657065e-07 -7.666193e-08 -3.615107e-08
## [12,]  1.768894e-07  2.381085e-07  1.500923e-07  1.031000e-07  8.411383e-08
## [13,] -2.490897e-08 -1.050847e-07 -7.112202e-08 -3.467764e-08 -2.399241e-08
## [14,] -1.779113e-07  5.694711e-06  3.509633e-06 -7.032931e-07 -9.517121e-07
## [15,] -4.984798e-06 -8.865390e-06 -5.811213e-06 -3.223422e-06 -2.566731e-06
## [16,] -2.673772e-06  2.142614e-05  1.051231e-05 -3.899810e-06 -4.279987e-06
## [17,] -1.194017e-05 -2.470218e-05 -1.359724e-05 -4.630767e-06 -3.365368e-06
## [18,] -4.986030e-07 -8.681542e-07 -4.245507e-07  2.690554e-08  4.264741e-08
## [19,] -6.540151e-07 -5.299646e-07 -3.244098e-07 -1.629766e-08  2.007096e-08
## [20,] -5.996019e-08  4.309626e-07 -5.906517e-08  6.385114e-08  6.041758e-08
## [21,]  3.332664e-07  3.222869e-06  2.161859e-06  8.547662e-08  4.712432e-08
## [22,] -1.207121e-07  1.863307e-06  1.537670e-06 -5.457491e-07 -1.083866e-07
## [23,]  7.874036e-08  8.073639e-08 -4.690461e-07 -1.079644e-06 -2.511051e-07
## [24,]  3.409923e-08  3.376781e-07  1.719163e-07 -9.732661e-08 -1.161144e-06
## [25,]  1.058333e-08  2.228266e-07  1.839569e-07  1.035881e-07 -2.143641e-07
## [26,]  2.970636e-07  3.226419e-07  2.004861e-07  1.672219e-07  1.460341e-07
## [27,]  3.337105e-07  3.894039e-07  2.248863e-07  1.500234e-07  1.257446e-07
## [28,]  1.151535e-07  3.152132e-07  2.740403e-07  2.677838e-07  2.056288e-07
## [29,]  4.060101e-08  6.078775e-08  1.530521e-08 -2.796077e-08 -2.246610e-08
## [30,]  7.297109e-07  3.065684e-06  1.684969e-06  4.217451e-07  2.706689e-07
## [31,] -2.367191e-06  1.559209e-06  9.157635e-07  3.867483e-07  2.872345e-07
## [32,] -3.129192e-06 -3.154680e-06  3.298074e-07  1.737232e-07  1.255851e-07
## [33,] -3.194215e-06 -1.199533e-05 -4.119222e-06  3.339195e-07  3.289519e-07
## [34,]  3.500857e-07 -3.934757e-06 -5.315420e-06 -2.426388e-06 -6.662100e-08
## [35,]  2.420886e-07  6.370627e-07 -2.250528e-06 -3.275202e-06 -5.378339e-07
## [36,]  2.000209e-07  4.391676e-07 -1.939050e-08 -5.721107e-07 -3.323526e-06
## [37,]  1.119440e-07  1.425603e-07  8.981100e-08 -1.566859e-08 -9.823947e-07
## [38,]  6.125177e-07  7.785550e-07  4.882032e-07  3.651781e-07  3.054907e-07
## [39,]  5.355741e-07  5.182179e-07  3.265015e-07  3.065681e-07  2.611856e-07
## [40,]  5.770584e-07  1.350364e-06  8.863995e-07  5.034411e-07  3.724199e-07
## [41,] -1.380047e-07 -3.780203e-07 -2.287294e-07 -8.278386e-08 -5.567269e-08
## [42,] -3.923758e-06 -6.986697e-06 -3.635719e-06 -3.320289e-07 -9.991132e-08
## [43,] -3.969371e-06 -2.287844e-06 -1.443921e-06 -6.277747e-08  9.280818e-08
## [44,]  4.333893e-07  2.666710e-06 -2.025274e-07  1.835398e-07  1.831940e-07
## [45,]  2.149683e-06  1.511125e-05  8.682975e-06 -4.109897e-08 -3.930105e-08
## [46,] -4.517727e-07  8.284322e-06  5.936469e-06 -2.129818e-06 -4.150470e-07
## [47,]  2.184847e-07  6.828654e-07 -1.499984e-06 -3.469751e-06 -7.196111e-07
## [48,]  2.134939e-07  6.417429e-07  1.371419e-07 -5.902032e-07 -4.237989e-06
## [49,]  1.350074e-07  3.451225e-07  2.638056e-07  9.950698e-08 -1.053039e-06
## [50,]  1.237830e-06  1.536720e-06  9.634531e-07  7.354653e-07  6.202978e-07
## [51,]  1.492431e-06  2.061324e-06  1.229349e-06  7.583240e-07  6.123478e-07
## [52,]  1.520648e-07  2.644276e-07  4.097710e-07  6.830317e-07  5.462364e-07
## [53,]  2.824935e-07  6.086507e-07  2.938259e-07 -1.409863e-08 -2.453321e-08
## [54,]  7.133470e-06  2.115522e-05  1.131217e-05  2.186001e-06  1.299628e-06
## [55,] -8.824615e-06  7.675919e-06  4.264992e-06  1.637419e-06  1.225352e-06
## [56,]  3.892645e-04 -1.098939e-05  1.159051e-06  6.584598e-07  4.736419e-07
## [57,] -1.098939e-05  3.539496e-04 -1.663391e-05  1.628115e-06  1.350920e-06
## [58,]  1.159051e-06 -1.663391e-05  3.816987e-04 -6.326124e-06  7.029554e-08
## [59,]  6.584598e-07  1.628115e-06 -6.326124e-06  3.908562e-04 -1.379895e-06
## [60,]  4.736419e-07  1.350920e-06  7.029554e-08 -1.379895e-06  3.913899e-04
## [61,]  3.043313e-07  6.346856e-07  4.497465e-07  1.145217e-07 -2.404668e-06
## [62,]  2.550531e-06  3.221291e-06  2.016550e-06  1.514337e-06  1.267976e-06
## [63,]  2.023689e-06  1.944704e-06  1.319331e-06  1.390837e-06  1.174849e-06
## [64,]  2.801987e-06  5.970080e-06  3.650048e-06  1.689025e-06  1.231372e-06
## [65,] -9.390133e-07 -2.094020e-06 -1.162824e-06 -2.845494e-07 -1.805160e-07
## [66,] -1.235441e-05  9.698944e-05  7.743364e-05  4.605987e-05  3.035929e-05
##               [,61]         [,62]         [,63]         [,64]         [,65]
##  [1,] -9.085531e-07 -4.706613e-07 -3.944244e-07 -1.870679e-06 -2.623744e-07
##  [2,]  3.640897e-08  3.231449e-07  5.704071e-07 -4.237973e-07  3.281596e-07
##  [3,]  3.188060e-08  3.295320e-07  2.721275e-07  2.945506e-07 -1.038818e-07
##  [4,]  4.302856e-09  2.645382e-07  2.190975e-07  1.477507e-07 -8.064689e-08
##  [5,]  1.141384e-09  4.228842e-07  3.272147e-07  2.671009e-07 -1.633992e-07
##  [6,]  1.001888e-08  2.937690e-07  2.270473e-07  2.132269e-07 -1.137058e-07
##  [7,] -4.279685e-09  2.454011e-07  2.190376e-07  1.846034e-07 -5.224511e-08
##  [8,] -4.881638e-07  2.658591e-07  2.298862e-07  5.532938e-08 -6.269253e-08
##  [9,] -3.024492e-07 -1.575453e-07  9.376540e-08 -3.538897e-08 -2.645135e-08
## [10,] -2.900360e-07 -1.871266e-06 -9.292369e-07 -2.251384e-07  7.301313e-08
## [11,] -3.439385e-08 -9.054774e-07 -1.834886e-06 -4.834754e-07 -4.107026e-07
## [12,]  3.891545e-08 -6.092530e-08 -2.615104e-07 -1.001870e-06  1.344665e-07
## [13,]  1.232394e-09  1.262812e-07 -3.498364e-07  1.484420e-07 -5.748620e-07
## [14,] -5.380796e-07 -4.556676e-06 -4.847940e-06 -2.461344e-06 -3.900477e-07
## [15,] -1.278984e-06 -5.646235e-06 -5.345072e-06 -4.269794e-06  3.013515e-07
## [16,] -2.815932e-06 -2.404563e-05 -2.819765e-05 -4.956157e-06 -4.255208e-06
## [17,] -2.423009e-06 -2.317509e-05 -1.902918e-05 -2.326748e-05  7.821312e-06
## [18,]  7.057593e-09 -1.407621e-07 -2.593843e-07  3.030272e-07 -1.506008e-07
## [19,]  3.425739e-08  4.931705e-07  4.419366e-07  3.387291e-07 -1.051856e-07
## [20,]  3.433293e-08  2.453924e-07  2.794607e-07  9.304241e-08  3.257159e-08
## [21,]  1.956202e-08  1.758668e-08  1.741404e-07 -2.830047e-07  2.265220e-07
## [22,]  1.261169e-08  1.143902e-07  2.063853e-07 -1.538405e-07  1.261149e-07
## [23,]  1.395958e-08  3.162738e-07  3.301654e-07  1.829941e-07 -1.009417e-09
## [24,] -2.387025e-07  2.277799e-07  2.360302e-07  2.776452e-07  4.000372e-09
## [25,] -1.273888e-07 -4.363286e-07  3.537076e-08  1.375789e-07  2.015995e-08
## [26,] -4.546384e-07 -3.638440e-06 -1.908258e-06 -3.526062e-07  1.772239e-07
## [27,]  6.225918e-09 -1.855416e-06 -3.518740e-06 -1.014410e-06 -5.788089e-07
## [28,]  7.687360e-08 -4.105649e-07 -1.078868e-06 -1.219132e-06 -3.265516e-08
## [29,]  1.336720e-08  3.240192e-07 -4.182323e-07  1.315142e-08 -7.311689e-07
## [30,]  1.796828e-07  1.572070e-06  2.802106e-06 -2.062801e-06  1.615359e-06
## [31,]  1.791907e-07  1.445762e-06  1.372137e-06  1.076204e-06 -2.147880e-07
## [32,]  8.185267e-08  6.949025e-07  5.560588e-07  7.440060e-07 -2.481778e-07
## [33,]  1.659190e-07  9.620267e-07  6.436762e-07  1.534086e-06 -5.314212e-07
## [34,]  1.246225e-07  6.312407e-07  4.401327e-07  1.004426e-06 -3.222827e-07
## [35,]  3.319196e-08  5.282328e-07  4.794553e-07  5.893129e-07 -1.062475e-07
## [36,] -9.543276e-07  4.706629e-07  4.262831e-07  4.235059e-07 -8.135638e-08
## [37,] -5.712252e-07 -8.098139e-07  1.019899e-07  1.614874e-07 -8.715509e-09
## [38,] -8.968506e-07 -7.530731e-06 -4.139729e-06 -7.535748e-07  3.990821e-07
## [39,]  1.927105e-08 -4.122257e-06 -7.424245e-06 -1.807791e-06 -1.170787e-06
## [40,]  1.550261e-07 -7.109106e-07 -1.736432e-06 -2.889274e-06  3.912310e-07
## [41,]  1.122520e-08  5.095831e-07 -1.052408e-06  4.231532e-07 -1.454784e-06
## [42,] -1.553443e-07 -2.218988e-06 -4.323876e-06  4.207943e-06 -2.775501e-06
## [43,]  1.426685e-07  1.982834e-06  1.346055e-06  2.410581e-06 -1.030463e-06
## [44,]  1.083333e-07  8.183503e-07  9.164065e-07  3.256870e-07  8.559059e-08
## [45,] -1.968500e-08 -3.068736e-07  3.164284e-07 -1.576356e-06  9.297150e-07
## [46,]  6.332221e-09  1.420608e-07  4.685126e-07 -7.379425e-07  4.660659e-07
## [47,]  4.028426e-08  8.261575e-07  8.487383e-07  6.156644e-07 -2.450515e-08
## [48,] -1.059868e-06  7.239407e-07  7.039471e-07  6.697034e-07 -5.605253e-08
## [49,] -7.061983e-07 -1.607439e-06  1.452341e-07  3.136289e-07  2.721466e-08
## [50,] -1.654830e-06 -1.515291e-05 -8.703126e-06 -1.618974e-06  8.884749e-07
## [51,]  1.232347e-07 -8.281730e-06 -1.425069e-05 -4.521949e-06 -1.453069e-06
## [52,]  1.959638e-07 -2.576817e-06 -5.676894e-06 -3.389323e-06 -5.865596e-07
## [53,]  7.428220e-08  1.578102e-06 -7.184947e-07 -3.433861e-07 -1.735594e-06
## [54,]  1.006008e-06  1.018009e-05  1.811769e-05 -1.389849e-05  1.039450e-05
## [55,]  8.327324e-07  7.418360e-06  7.931048e-06  3.055636e-06  1.550522e-07
## [56,]  3.043313e-07  2.550531e-06  2.023689e-06  2.801987e-06 -9.390133e-07
## [57,]  6.346856e-07  3.221291e-06  1.944704e-06  5.970080e-06 -2.094020e-06
## [58,]  4.497465e-07  2.016550e-06  1.319331e-06  3.650048e-06 -1.162824e-06
## [59,]  1.145217e-07  1.514337e-06  1.390837e-06  1.689025e-06 -2.845494e-07
## [60,] -2.404668e-06  1.267976e-06  1.174849e-06  1.231372e-06 -1.805160e-07
## [61,]  3.983830e-04 -3.120332e-06  2.427434e-07  5.708507e-07  1.984397e-08
## [62,] -3.120332e-06  3.690023e-04 -1.860560e-05 -3.505100e-06  1.981864e-06
## [63,]  2.427434e-07 -1.860560e-05  3.689478e-04 -7.583911e-06 -3.460512e-06
## [64,]  5.708507e-07 -3.505100e-06 -7.583911e-06  3.897585e-04  1.665966e-06
## [65,]  1.984397e-08  1.981864e-06 -3.460512e-06  1.665966e-06  3.954011e-04
## [66,]  9.251479e-06 -1.996044e-05 -1.763500e-05  3.001049e-05  4.189432e-06
##               [,66]
##  [1,] -1.260489e-04
##  [2,] -3.626851e-06
##  [3,] -6.310892e-06
##  [4,] -1.492951e-05
##  [5,] -2.623075e-05
##  [6,] -1.526080e-05
##  [7,] -4.299249e-06
##  [8,] -2.218816e-05
##  [9,] -1.939079e-05
## [10,] -1.241507e-05
## [11,] -1.998256e-05
## [12,] -1.396871e-06
## [13,] -2.320301e-06
## [14,]  5.702419e-05
## [15,]  8.021419e-05
## [16,]  9.057870e-05
## [17,]  1.570613e-04
## [18,]  1.018219e-05
## [19,] -1.238544e-05
## [20,]  1.297754e-06
## [21,]  7.913297e-06
## [22,] -1.554400e-06
## [23,]  9.980407e-07
## [24,]  1.880316e-05
## [25,]  1.202925e-05
## [26,] -3.571892e-06
## [27,] -6.210078e-06
## [28,]  1.501497e-05
## [29,] -3.646624e-06
## [30,] -8.113671e-06
## [31,] -3.140865e-06
## [32,] -4.359057e-06
## [33,]  1.756284e-05
## [34,]  1.582217e-05
## [35,]  1.449993e-05
## [36,]  4.451148e-06
## [37,] -2.014438e-06
## [38,] -4.765784e-06
## [39,] -8.077459e-06
## [40,]  1.737217e-05
## [41,] -3.046768e-06
## [42,]  5.024874e-05
## [43,] -4.958148e-05
## [44,]  2.047514e-06
## [45,]  9.200804e-06
## [46,] -7.464115e-06
## [47,]  1.460847e-05
## [48,]  2.248193e-05
## [49,]  9.805632e-06
## [50,] -9.925606e-06
## [51,] -1.594710e-05
## [52,]  3.626311e-05
## [53,] -8.524880e-06
## [54,] -1.145423e-04
## [55,] -4.958512e-05
## [56,] -1.235441e-05
## [57,]  9.698944e-05
## [58,]  7.743364e-05
## [59,]  4.605987e-05
## [60,]  3.035929e-05
## [61,]  9.251479e-06
## [62,] -1.996044e-05
## [63,] -1.763500e-05
## [64,]  3.001049e-05
## [65,]  4.189432e-06
## [66,]  5.401626e-03
## 
## $log_evidence
## [1] -131.9296
## 
## $converge
## [1] "YES"
## 
## $iter_counts
## [1] 211

3i)

Use the viz_post_coefs() function to visualize the posterior coefficient summaries for model 3 and model 6, based on the very strong prior specification.

SOLUTION

### add more code chunks if you like
viz_post_coefs(laplace_03_very_strong$mode[-length(laplace_03_very_strong$mode)],
               sqrt(diag(laplace_03_very_strong$var_matrix))[-length(laplace_03_very_strong$mode)],
               info_03_very_strong$design_matrix %>% colnames())

viz_post_coefs(laplace_06_very_strong$mode[-length(laplace_06_very_strong$mode)],
               sqrt(diag(laplace_06_very_strong$var_matrix))[-length(laplace_06_very_strong$mode)],
               info_06_very_strong$design_matrix %>% colnames())

3j)

Describe the influence of the regression coefficient prior standard deviation on the coefficient posterior distributions.

SOLUTION

What do you think?

As the prior becomes stronger, our posterior means become more and more confined around the prior. We can see this in scale of the distribution of the coefficient value become smaller and smaller.

3k)

You previously compared the two models using the Bayes Factor based on the weak prior specification.

Compare the performance of the two models with Bayes Factors again, but considering the results based on the strong and very strong priors. Does the prior influence which model is considered to be better?

SOLUTIOn

### add more code chunks if you like
log_evidences <- c(laplace_03_strong$log_evidence, laplace_06_strong$log_evidence)
if (exp(log_evidences[1]) > exp(log_evidences[2])) {
  sprintf("Model 3 is better and has bayes factor of:%f", exp(log_evidences[1])/sum(exp(log_evidences)))
} else {
  sprintf("Model 6 is better and has bayes factor of:%f", exp(log_evidences[2])/sum(exp(log_evidences)))
}

## [1] "Model 3 is better and has bayes factor of:1.000000"

log_evidences <- c(laplace_03_very_strong$log_evidence, laplace_06_very_strong$log_evidence)
if (exp(log_evidences[1]) > exp(log_evidences[2])) {
  sprintf("Model 3 is better and has bayes factor of:%f", exp(log_evidences[1])/sum(exp(log_evidences)))
} else {
  sprintf("Model 6 is better and has bayes factor of:%f", exp(log_evidences[2])/sum(exp(log_evidences)))
}

## [1] "Model 6 is better and has bayes factor of:0.999756"

Yes, clearly the prior has influence on the results of the bayes factor.

Problem 04

You examined the behavior of the coefficient posterior based on the influence of the prior. Let’s now consider the prior’s influence by examining the posterior predictive distributions.

4a)

You will make posterior predictions following the approach from the previous assignment. Posterior samples are generated and those samples are used to calculate the posterior samples of the mean trend and generate random posterior samples of the response around the mean. In the previous assignment, you made posterior predictions in order to calculate errors. In this assignment, you will not calculate errors, instead you will summarize the posterior predictions of the mean and of the random response.

The generate_lm_post_samples() function is defined for you below. It uses the MASS::mvrnorm() function generate posterior samples from the Laplace Approximation’s MVN distribution.

generate_lm_post_samples <- function(mvn_result, length_beta, num_samples)
{
  MASS::mvrnorm(n = num_samples,
                mu = mvn_result$mode,
                Sigma = mvn_result$var_matrix) %>% 
    as.data.frame() %>% tibble::as_tibble() %>% 
    purrr::set_names(c(sprintf("beta_%02d", 0:(length_beta-1)), "varphi")) %>% 
    mutate(sigma = exp(varphi))
}

The code chunk below starts the post_lm_pred_samples() function. This function generates posterior mean trend predictions and posterior predictions of the response. The first argument, Xnew, is a potentially new or test design matrix that we wish to make predictions at. The second argument, Bmat, is a matrix of posterior samples of the \(\boldsymbol{\beta}\)-parameters, and the third argument, sigma_vector, is a vector of posterior samples of the likelihood noise. The Xnew matrix has rows equal to the number of predictions points, M, and the Bmat matrix has rows equal to the number of posterior samples S.

You must complete the function by performing the necessary matrix math to calculate the matrix of posterior mean trend predictions, Umat, and the matrix of posterior response predictions, Ymat. You must also complete missing arguments to the definition of the Rmat and Zmat matrices. The Rmat matrix replicates the posterior likelihood noise samples the correct number of times. The Zmat matrix is the matrix of randomly generated standard normal values. You must correctly specify the required number of rows to the Rmat and Zmat matrices.

The post_lm_pred_samples() returns the Umat and Ymat matrices contained within a list.

Perform the necessary matrix math to calculate the matrix of posterior predicted mean trends Umat and posterior predicted responses Ymat. You must specify the number of required rows to create the Rmat and Zmat matrices.

HINT: The following code chunk should look famaliar…

SOLUTION

post_lm_pred_samples <- function(Xnew, Bmat, sigma_vector)
{
  # number of new prediction locations
  M <- nrow(Xnew)
  # number of posterior samples
  S <- nrow(Bmat)
  
  # matrix of linear predictors
  Umat <- Xnew %*% t(Bmat)
  
  # assmeble matrix of sigma samples, set the number of rows
  Rmat <- matrix(rep(sigma_vector, M), M, byrow = TRUE)
  
  # generate standard normal and assemble into matrix
  # set the number of rows
  Zmat <- matrix(rnorm(M*S), M, byrow = TRUE)
  
  # calculate the random observation predictions
  Ymat <- Umat + Rmat * Zmat
  
  # package together
  list(Umat = Umat, Ymat = Ymat)
}

4b)

Since this assignment is focused on visualizing the predictions, we will summarize the posterior predictions to focus on the posterior means and the middle 95% uncertainty intervals. The code chunk below is defined for you which serves as a useful wrapper function to call post_lm_pred_samples().

make_post_lm_pred <- function(Xnew, post)
{
  Bmat <- post %>% select(starts_with("beta_")) %>% as.matrix()
  
  sigma_vector <- post %>% pull(sigma)
  
  post_lm_pred_samples(Xnew, Bmat, sigma_vector)
}

The code chunk below defines a function summarize_lm_pred_from_laplace() which manages the actions necessary to summarize posterior predictions. The first argument, mvn_result, is the Laplace Approximation object. The second object is the test design matrix, Xtest, and the third argument, num_samples, is the number of posterior samples to make.

You must complete the code chunk below which summarizes the posterior predictions. This function takes care of most of the coding for you. You do not have to worry about the generation of the posterior samples OR calculating the posterior quantiles associated with the middle 95% uncertainty interval. You must calculate the posterior average by deciding on whether you should use colMeans() or rowMeans() to calculate the average across all posterior samples per prediction location.

Follow the comments in the code chunk below to complete the definition of the summarize_lm_pred_from_laplace() function. You must calculate the average posterior mean trend and the average posterior response.

SOLUTION

summarize_lm_pred_from_laplace <- function(mvn_result, Xtest, num_samples)
{
  # generate posterior samples of the beta parameters
  post <- generate_lm_post_samples(mvn_result, ncol(Xtest), num_samples)
  
  # make posterior predictions on the test set
  pred_test <- make_post_lm_pred(Xtest, post)
  
  # calculate summary statistics on the predicted mean and response
  # summarize over the posterior samples
  
  # posterior mean, should you summarize along rows (rowMeans) or 
  # summarize down columns (colMeans) ???
  mu_avg <- rowMeans(pred_test$Umat)
  y_avg <- rowMeans(pred_test$Ymat)
  
  # posterior quantiles for the middle 95% uncertainty intervals
  mu_lwr <- apply(pred_test$Umat, 1, stats::quantile, probs = 0.025)
  mu_upr <- apply(pred_test$Umat, 1, stats::quantile, probs = 0.975)
  y_lwr <- apply(pred_test$Ymat, 1, stats::quantile, probs = 0.025)
  y_upr <- apply(pred_test$Ymat, 1, stats::quantile, probs = 0.975)
  
  # book keeping
  tibble::tibble(
    mu_avg = mu_avg,
    mu_lwr = mu_lwr,
    mu_upr = mu_upr,
    y_avg = y_avg,
    y_lwr = y_lwr,
    y_upr = y_upr
  ) %>% 
    tibble::rowid_to_column("pred_id")
}

4c)

When you made predictions in Problem 02, the lm() object handled making the test design matrix. However, since we have programmed the Bayesian modeling approach from scratch we need to create the test design matrix manually.

Create the test design matrix based on the visualization grid, viz_grid, using the model 3 formulation. Assign the result to the X03_test object.

#y ~ (x1 + I(x1^2))*(x2 + I(x2^2)
x03_test <- model.matrix(~ (x1 + I(x1^2))*(x2 + I(x2^2)), data = viz_grid)

Call the summarize_lm_pred_from_laplace() function to summarize the posterior predictions from the model 3 formulation for the weak, strong, and very strong prior specifications. Use 5000 posterior samples for each case. Assign the results from the weak prior to post_pred_summary_viz_03_weak, the results from the strong prior to post_pred_summary_viz_03_strong, and the results from the very strong prior to post_pred_summary_viz_03_very_strong.

SOLUTION

### add as many code chunks as you'd like
post_pred_summary_viz_03_very_strong <- summarize_lm_pred_from_laplace(laplace_03_very_strong, x03_test, 5000)
post_pred_summary_viz_03_strong <- summarize_lm_pred_from_laplace(laplace_03_strong, x03_test, 5000)
post_pred_summary_viz_03_weak <- summarize_lm_pred_from_laplace(laplace_03_weak, x03_test, 5000)

4d)

You will now visualize the posterior predictions from the model 3 Bayesian models associated with the weak, strong, and very strong priors. The viz_grid object is joined to the prediction dataframes assuming you have used the correct variable names!

Visualize the predicted means, confidence intervals, and prediction intervals in the style of those that you created in Problem 02. The confidence interval bounds are mu_lwr and mu_upr columns and the prediction interval bounds are the y_lwr and y_upr columns, respectively. The posterior predicted mean of the mean is mu_avg.

Pipe the result of the joined dataframe into ggplot() and make appropriate aesthetics and layers to visualize the predictions with the x1 variable mapped to the x aesthetic and the x2 variable used as a facet variable.

SOLUTION

post_pred_summary_viz_03_weak %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id') %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) + 
  facet_wrap(~ x2)

post_pred_summary_viz_03_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id') %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) + 
  facet_wrap(~ x2)

post_pred_summary_viz_03_very_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id') %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) + 
  facet_wrap(~ x2)

4e)

In order to make posterior predictions for the model 6 formulation you must create a test design matrix consistent with the training set basis. The code chunk below creates a helper function which extracts the interior and boundary knots of a natural spline associated with the training set for you. The first argument, J, is the degrees-of-freedom (DOF) of the spline, the second argument, train_data, is the training data set. The third argument xname is the name of the variable you are applying the spline to. The xname argument must be provided as a character string.

make_splines_training_knots <- function(J, train_data, xname)
{
  # extract the input from the training set
  x <- train_data %>% select(all_of(xname)) %>% pull()
  
  # create the training basis
  train_basis <- splines::ns(x, df = J)
  
  # extract the knots
  interior_knots <- as.vector(attributes(train_basis)$knots)
  
  boundary_knots <- as.vector(attributes(train_basis)$Boundary.knots)
  
  # book keeping
  list(interior_knots = interior_knots,
       boundary_knots = boundary_knots)
}

Create the test design matrix based on the visualization grid, viz_grid, using the model 6 formulation. Assign the result to the X06_test object. Use the make_splines_training_knots() function to get the interior and boundary knots associated with the training set for the x1 variable to create the test design matrix.

#?all_of
knot_info <- make_splines_training_knots(12, viz_grid, c("x1"))
X06_test = model.matrix(
  ~ splines::ns(x1,
                knots = knot_info$interior_knots,
                Boundary.knots = knot_info$boundary_knots) * 
    (x2 +I(x2^2) + I(x2^3) + I(x2^4)), 
  data = viz_grid)

Call the summarize_lm_pred_from_laplace() function to summarize the posterior predictions from the model 6 formulation for the weak, strong, and very strong prior specifications. Use 5000 posterior samples for each case. Assign the results from the weak prior to post_pred_summary_viz_06_weak, the results from the strong prior to post_pred_summary_viz_06_strong, and the results from the very strong prior to post_pred_summary_viz_06_very_strong.

HINT: The make_spline_training_knots() function returns a list! The fields or elements of the list can be accessed via the $ operator.

SOLUTION

### add as many code chunks as you'd like
post_pred_summary_viz_06_weak <- summarize_lm_pred_from_laplace(laplace_06_weak, X06_test, 5000)
post_pred_summary_viz_06_strong <- summarize_lm_pred_from_laplace(laplace_06_strong, X06_test, 5000)
post_pred_summary_viz_06_very_strong <- summarize_lm_pred_from_laplace(laplace_06_very_strong, X06_test, 5000)

4f)

You will now visualize the posterior predictions from the model 6 Bayesian models associated with the weak, strong, and very strong priors. The viz_grid object is joined to the prediction dataframes assuming you have used the correct variable names!

Visualize the predicted means, confidence intervals, and prediction intervals in the style of those that you created in Problem 02. The confidence interval bounds are mu_lwr and mu_upr columns and the prediction interval bounds are the y_lwr and y_upr columns, respectively. The posterior predicted mean of the mean is mu_avg.

Pipe the result of the joined dataframe into ggplot() and make appropriate aesthetics and layers to visualize the predictions with the x1 variable mapped to the x aesthetic and the x2 variable used as a facet variable.

SOLUTION

post_pred_summary_viz_06_weak %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id') %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) + 
  facet_wrap(~ x2)

post_pred_summary_viz_06_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id') %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) + 
  facet_wrap(~ x2)

post_pred_summary_viz_06_very_strong %>% 
  left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
            by = 'pred_id') %>%
  ggplot(mapping = aes(x = x1)) +
  geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
  geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
  geom_line(mapping = aes(y = mu_avg)) + 
  facet_wrap(~ x2)

4g)

Describe the behavior of the predictions as the prior standard deviation decreased. Are the posterior predictions consistent with the behavior of the posterior coefficients?

SOLUTION

What do you think?

The confidence intervals get tighter with increase in strength of the prior. Yes, as the means are more confined, so is the mean trend of the posterior.

Problem 05

Now that you have worked with Bayesian models with the prior regularizing the coefficients, you will consider non-Bayesian regularization methods. You will work with the glmnet package in this problem which takes care of all fitting and visualization for you.

The code chunk below loads in glmnet and so you must have glmnet installed before running this code chunk. IMPORANT: the eval flag is set to FALSE below. Once you download glmnet set eval=TRUE.

library(glmnet)

## Loading required package: Matrix

## 
## Attaching package: 'Matrix'

## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack

## Loaded glmnet 4.1-8

5a)

glmnet does not work with the formula interface. And so you must create the training design matrix. However, glmnet prefers the the intercept column of ones to not be included in the design matrix. To support that you must define new design matrices. These matrices will use the same formulation but you must remove the intercept column. This is easy to do with the formula interface and the model.matrix() function. Include - 1 in the formula and model.matrix() will not include the intercept. The code chunk below demonstrates removing the intercept column for a model with linear additive features.

model.matrix( y ~ x1 + x2 - 1, data = df) %>% head()

##           x1         x2
## 1 -0.3092328  0.3087799
## 2  0.6312721 -0.5479198
## 3 -0.6827669  2.1664494
## 4  0.2693056  1.2097037
## 5  0.3725202  0.7854860
## 6  1.2966439 -0.1877231

Create the design matrices for glmnet for the model 3 and model 6 formulations. Remove the intercept column for both and assign the results to X03_glmnet and X06_glmnet.

SOLUTION

### add more code chunks if you prefer

X03_glmnet <- model.matrix(y ~ (x1 + I(x1^2))*(x2 + I(x2^2) - 1), data = df)
X06_glmnet <- model.matrix(y ~ splines::ns(x1, df = 12) * (x2 +I(x2^2) + I(x2^3) + I(x2^4)) - 1, data = df)

5b)

By default glmnet uses the lasso penalty. Fit a Lasso model by calling glmnet(). The first argument to glmnet() is the design matrix and the second argument is a regular vector for the response.

Train a Lasso model for the model 3 and model 6 formulations, assign the results to lasso_03 and lasso_06, respectively.

SOLUTION

### add more code chunks if you like
#?glmnet
lasso_03 <- glmnet(X03_glmnet, df$y)
lasso_06 <- glmnet(X06_glmnet, df$y)

5c)

Plot the coefficient path for each Lasso model by calling the plot() function on the glmnet model object. Specify the xvar argument to be 'lambda' in the plot() call.

SOLUTION

### add more code chunks if you like
lasso_03 %>% plot(xvar = 'lambda')

lasso_06 %>% plot(xvar = 'lambda')

5d)

Now that you have visualized the coefficient path, it’s time to identify the best 'lambda' value to use! The cv.glmnet() function will by default use 10-fold cross-validation to tune 'lambda'. The first argument to cv.glmnet() is the design matrix and the second argument is the regular vector for the response.

Tune the Lasso regularization strength with cross-validation using the cv.glmnet() function for each model formulation. Assign the model 3 result to lasso_03_cv_tune and assign the model 6 result to lasso_06_cv_tune. Also specify the alpha argument to be 1 to make sure the Lasso penalty is applied in the cv.glmnet() call.

HINT: The random seed was assigned for you in two separate code chunks below. This will help ensure you can reproduce the cross-validation results.

SOLUTION

### the random seed is set for you
set.seed(812312)
### tune the model 3 formulation
lasso_03_cv_tune <- cv.glmnet(X03_glmnet, df$y, alpha = 1)

### the random seed is set for you
set.seed(812312)
### tune the model 6 formulation
lasso_06_cv_tune <- cv.glmnet(X06_glmnet, df$y, alpha = 1)

5e)

Plot the cross-validation results using the default plot method for each cross-validation result. How many coefficients are remaining after tuning?

SOLUTION

### add more code chunks if you like
lasso_03_cv_tune %>% plot()

lasso_06_cv_tune %>% plot()

5f)

Which features have NOT been “turned off” by the Lasso penalty? Use the coef() function to display the lasso model cross-validation results to show the tuned penalized regression coefficients for each model.
Are the final tuned models different from each other?

SOLUTION

### add more code chunks if you like
lasso_03_cv_tune %>% coef()

## 9 x 1 sparse Matrix of class "dgCMatrix"
##                         s1
## (Intercept)      0.3009999
## x1               .        
## I(x1^2)          .        
## x2               .        
## I(x2^2)         -0.3040403
## x1:x2            .        
## x1:I(x2^2)       .        
## I(x1^2):x2       .        
## I(x1^2):I(x2^2)  .

?coef
lasso_06_cv_tune %>% coef()

## 65 x 1 sparse Matrix of class "dgCMatrix"
##                                            s1
## (Intercept)                         0.2818858
## splines::ns(x1, df = 12)1           .        
## splines::ns(x1, df = 12)2           .        
## splines::ns(x1, df = 12)3           .        
## splines::ns(x1, df = 12)4           .        
## splines::ns(x1, df = 12)5           .        
## splines::ns(x1, df = 12)6           .        
## splines::ns(x1, df = 12)7           .        
## splines::ns(x1, df = 12)8           .        
## splines::ns(x1, df = 12)9           .        
## splines::ns(x1, df = 12)10          .        
## splines::ns(x1, df = 12)11          .        
## splines::ns(x1, df = 12)12          .        
## x2                                  .        
## I(x2^2)                            -0.2847331
## I(x2^3)                             .        
## I(x2^4)                             .        
## splines::ns(x1, df = 12)1:x2        .        
## splines::ns(x1, df = 12)2:x2        .        
## splines::ns(x1, df = 12)3:x2        .        
## splines::ns(x1, df = 12)4:x2        .        
## splines::ns(x1, df = 12)5:x2        .        
## splines::ns(x1, df = 12)6:x2        .        
## splines::ns(x1, df = 12)7:x2        .        
## splines::ns(x1, df = 12)8:x2        .        
## splines::ns(x1, df = 12)9:x2        .        
## splines::ns(x1, df = 12)10:x2       .        
## splines::ns(x1, df = 12)11:x2       .        
## splines::ns(x1, df = 12)12:x2       .        
## splines::ns(x1, df = 12)1:I(x2^2)   .        
## splines::ns(x1, df = 12)2:I(x2^2)   .        
## splines::ns(x1, df = 12)3:I(x2^2)   .        
## splines::ns(x1, df = 12)4:I(x2^2)   .        
## splines::ns(x1, df = 12)5:I(x2^2)   .        
## splines::ns(x1, df = 12)6:I(x2^2)   .        
## splines::ns(x1, df = 12)7:I(x2^2)   .        
## splines::ns(x1, df = 12)8:I(x2^2)   .        
## splines::ns(x1, df = 12)9:I(x2^2)   .        
## splines::ns(x1, df = 12)10:I(x2^2)  .        
## splines::ns(x1, df = 12)11:I(x2^2)  .        
## splines::ns(x1, df = 12)12:I(x2^2)  .        
## splines::ns(x1, df = 12)1:I(x2^3)   .        
## splines::ns(x1, df = 12)2:I(x2^3)   .        
## splines::ns(x1, df = 12)3:I(x2^3)   .        
## splines::ns(x1, df = 12)4:I(x2^3)   .        
## splines::ns(x1, df = 12)5:I(x2^3)   .        
## splines::ns(x1, df = 12)6:I(x2^3)   .        
## splines::ns(x1, df = 12)7:I(x2^3)   .        
## splines::ns(x1, df = 12)8:I(x2^3)   .        
## splines::ns(x1, df = 12)9:I(x2^3)   .        
## splines::ns(x1, df = 12)10:I(x2^3)  .        
## splines::ns(x1, df = 12)11:I(x2^3)  .        
## splines::ns(x1, df = 12)12:I(x2^3)  .        
## splines::ns(x1, df = 12)1:I(x2^4)   .        
## splines::ns(x1, df = 12)2:I(x2^4)   .        
## splines::ns(x1, df = 12)3:I(x2^4)   .        
## splines::ns(x1, df = 12)4:I(x2^4)   .        
## splines::ns(x1, df = 12)5:I(x2^4)   .        
## splines::ns(x1, df = 12)6:I(x2^4)   .        
## splines::ns(x1, df = 12)7:I(x2^4)   .        
## splines::ns(x1, df = 12)8:I(x2^4)   .        
## splines::ns(x1, df = 12)9:I(x2^4)   .        
## splines::ns(x1, df = 12)10:I(x2^4)  .        
## splines::ns(x1, df = 12)11:I(x2^4)  .        
## splines::ns(x1, df = 12)12:I(x2^4)  .